%------------------------------------------------------------------------------
% File     : ITP285_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00107_006275
% 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    : 0094_VEBT_SuccPredImperative_00107_006275 [Des22]

% Status   : Theorem
% Rating   : 0.89 v8.2.0, 0.90 v8.1.0
% Syntax   : Number of formulae    : 16453 (3565 unt;5158 typ;   0 def)
%            Number of atoms       : 30444 (11212 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 22514 (3365   ~; 520   |;2255   &)
%                                         (2766 <=>;13608  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   39 (   2 avg)
%            Number of types       : 1199 (1198 usr)
%            Number of type conns  : 5737 (3376   >;2361   *;   0   +;   0  <<)
%            Number of predicates  :  152 ( 149 usr;   3 prp; 0-4 aty)
%            Number of functors    : 3811 (3811 usr; 583 con; 0-4 aty)
%            Number of variables   : 31524 (30497   !;1027   ?;31524   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 19:47:47.654
%------------------------------------------------------------------------------
% Could-be-implicit typings (1198)
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tff(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    set_Extended_enat: $tType ).

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

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

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

tff(ty_n_t__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__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
    fun_bool_bool: $tType ).

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

tff(ty_n_t__List__Olist_It__Assertions__Oassn_J,type,
    list_assn: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__HOL__Obool_J,type,
    fun_rat_bool: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
    option_real: $tType ).

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

tff(ty_n_t__Option__Ooption_It__HOL__Obool_J,type,
    option_bool: $tType ).

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

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

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

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

tff(ty_n_t__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__Heap__Oarray_It__HOL__Obool_J,type,
    array_bool: $tType ).

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

tff(ty_n_t__Heap__Oarray_It__Nat__Onat_J,type,
    array_nat: $tType ).

tff(ty_n_t__Heap__Oarray_It__Int__Oint_J,type,
    array_int: $tType ).

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

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__VEBT____Definitions__OVEBT,type,
    vEBT_VEBT: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

% Explicit typings (3960)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: fun_option_nat_assn ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: set_int > fun_bool_set_int ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: set_real > fun_nat_set_real ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: set_bool > fun_nat_set_bool ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: set_int > fun_nat_set_int ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: set_real > fun_int_set_real ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: set_set_VEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: set_set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: set_set_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: set_set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: set_set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: set_set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: set_set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_nat_set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_na6128946301132549280et_nat > fun_se3395149180098744406et_nat ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: fun_na6128946301132549280et_nat > fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: fun_na6128946301132549280et_nat > fun_se3395149180098744406et_nat ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: fun_nat_set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: fun_nat_set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: fun_int_set_nat > fun_fu5340233072860297857et_nat ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: fun_real_set_nat > fun_fu2450431469134963913et_nat ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_set_nat_set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: fun_int_set_nat > fun_fu173695997200249801et_nat ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_nat_set_nat > fun_fu5004605035024130925et_nat ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: fun_nat_real > fun_fu8407118354944431103t_real ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: fun_nat_set_nat > fun_fu2478289658772682457et_nat ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: fun_nat_int > fun_fu5355111151935471743et_int ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: fun_bool_assn > fun_as6968651184343528707l_assn ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_nat_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: int > fun_fu4581781630362247899et_int ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: bool > fun_fu1091034382302749763t_bool ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: vEBT_VEBT > fun_fu5930593133619422147T_VEBT ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: int > fun_fu4110326366348883971et_int ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: bool > fun_fu6333760075547975817t_bool ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: real > fun_fu4751776603652232707t_real ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: fun_nat_set_nat > fun_se3395149180098744406et_nat ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_nat_set_nat > fun_set_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: set_nat > fun_fu6967811735264326619et_nat ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: set_nat > fun_fu5244020444916840515et_nat ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: set_nat > fun_fu2074493022567586523et_nat ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: fun_real_set_nat > fun_se9064950638549745330et_nat ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_bool_set_nat > fun_se8186231001398342275et_nat ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: fun_nat_set_nat > fun_se3395149180098744406et_nat ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: fun_int_set_nat > fun_se180586794239543474et_nat ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: fun_nat_set_nat > fun_nat_set_set_nat ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: fun_nat_real > fun_nat_set_real ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: fun_nat_int > fun_nat_set_int ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: fun_int_int > fun_int_set_int ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: fun_nat_nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: set_op4508134149509766951at_nat > fun_op2627489687811228755t_bool ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: set_option_nat > fun_option_nat_bool ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: set_option_num > fun_option_num_bool ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: fun_VE3114949619472531401T_VEBT > fun_VE3640887651371881891T_VEBT ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: fun_na8123536816256238275T_VEBT > fun_na7453361831206303133T_VEBT ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: fun_re8294352169103142759T_VEBT > fun_re4523157989745374273T_VEBT ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: fun_bo1245769998338026534T_VEBT > fun_bo1266566265771860096T_VEBT ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_VE8613710830879766659t_real > fun_VE2077312251927228713t_real ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: fun_nat_list_real > fun_nat_set_real ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: fun_real_list_real > fun_real_set_real ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: fun_bool_list_real > fun_bool_set_real ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: fun_VE3624423849788874412t_bool > fun_VE4526288134577429638t_bool ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: fun_nat_list_bool > fun_nat_set_bool ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: set_complex > fun_set_complex_bool ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: set_Code_integer > fun_se2712834123562403997r_bool ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: set_int > fun_set_int_bool ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_VEBT_VEBTi_rat > fun_ra5096748620215079228at_rat ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_VEBT_VEBT_rat > fun_ra5804646369735708275at_rat ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_real_rat > fun_ra8745625228414021939at_rat ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: fun_bool_rat > fun_ra6264997190470632278at_rat ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: fun_nat_rat > fun_ra7317100687414072407at_rat ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: fun_int_rat > fun_ra1107035566227760819at_rat ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: fun_VEBT_VEBTi_int > fun_in5621419589014656484at_int ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: fun_VEBT_VEBT_int > fun_in3204970175496335219at_int ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: fun_real_int > fun_in8450036337100091355at_int ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: fun_nat_int > fun_in2640216123419753727at_int ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: code_integer > fun_fu3438678930028006669omplex ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: code_integer > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: code_integer > fun_fu3438678930028006669omplex ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: code_integer > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: fun_VEBT_VEBT_real > fun_VE8613710830879766659t_real ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: fun_Pr4298194452408323251l_bool > fun_Pr5830346466234525497t_bool ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: fun_nat_bool > fun_nat_list_bool ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: fun_VEBT_VEBT_nat > fun_VE4564399238913696295st_nat ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: fun_nat_nat > fun_nat_list_nat ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: fun_co7673397065115134084omplex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: fun_int_fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: fun_VEBT_VEBT_real > fun_li6194437398966043808t_real ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: fun_Pr4298194452408323251l_bool > fun_li7437436280209314921t_bool ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_bool_assn > fun_as6968651184343528707l_assn ).

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: fun_nat_bool > fun_li5572529915613225996t_bool ).

tff(sy_c_ATP_058Lamm__afb____,type,
    aTP_Lamm_afb: fun_VEBT_VEBT_nat > fun_li4186102260289925152st_nat ).

tff(sy_c_ATP_058Lamm__afc____,type,
    aTP_Lamm_afc: fun_nat_nat > fun_li8338050668395447434st_nat ).

tff(sy_c_ATP_058Lamm__afd____,type,
    aTP_Lamm_afd: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__afe____,type,
    aTP_Lamm_afe: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__aff____,type,
    aTP_Lamm_aff: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__afg____,type,
    aTP_Lamm_afg: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__afh____,type,
    aTP_Lamm_afh: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__afi____,type,
    aTP_Lamm_afi: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__afj____,type,
    aTP_Lamm_afj: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__afk____,type,
    aTP_Lamm_afk: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__afl____,type,
    aTP_Lamm_afl: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__afm____,type,
    aTP_Lamm_afm: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__afn____,type,
    aTP_Lamm_afn: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__afo____,type,
    aTP_Lamm_afo: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__afp____,type,
    aTP_Lamm_afp: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__afq____,type,
    aTP_Lamm_afq: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__afr____,type,
    aTP_Lamm_afr: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__afs____,type,
    aTP_Lamm_afs: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__aft____,type,
    aTP_Lamm_aft: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__afu____,type,
    aTP_Lamm_afu: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__afv____,type,
    aTP_Lamm_afv: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__afw____,type,
    aTP_Lamm_afw: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__afx____,type,
    aTP_Lamm_afx: fun_complex_complex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__afy____,type,
    aTP_Lamm_afy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__afz____,type,
    aTP_Lamm_afz: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__aga____,type,
    aTP_Lamm_aga: set_nat > fun_fu6002750694894812635t_bool ).

tff(sy_c_ATP_058Lamm__agb____,type,
    aTP_Lamm_agb: set_nat > fun_fu6772184894847113056BT_nat ).

tff(sy_c_ATP_058Lamm__agc____,type,
    aTP_Lamm_agc: fun_VE597728521808591204at_nat > fun_na6208021027755453088BT_nat ).

tff(sy_c_ATP_058Lamm__agd____,type,
    aTP_Lamm_agd: set_VEBT_VEBT > fun_fu4406026827936152384at_nat ).

tff(sy_c_ATP_058Lamm__age____,type,
    aTP_Lamm_age: set_nat > fun_fu9073671928554219267t_bool ).

tff(sy_c_ATP_058Lamm__agf____,type,
    aTP_Lamm_agf: set_nat > fun_fu6463170612142064108al_nat ).

tff(sy_c_ATP_058Lamm__agg____,type,
    aTP_Lamm_agg: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__agh____,type,
    aTP_Lamm_agh: set_real > fun_fu3370445115705697296at_nat ).

tff(sy_c_ATP_058Lamm__agi____,type,
    aTP_Lamm_agi: set_nat > fun_fu2583778531842586371t_bool ).

tff(sy_c_ATP_058Lamm__agj____,type,
    aTP_Lamm_agj: set_nat > fun_fu8221675813083619692nt_nat ).

tff(sy_c_ATP_058Lamm__agk____,type,
    aTP_Lamm_agk: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__agl____,type,
    aTP_Lamm_agl: set_int > fun_fu7719728765329718531t_bool ).

tff(sy_c_ATP_058Lamm__agm____,type,
    aTP_Lamm_agm: set_int > fun_fu5002972218919249680at_nat ).

tff(sy_c_ATP_058Lamm__agn____,type,
    aTP_Lamm_agn: set_nat > fun_fu5088574307552523139t_bool ).

tff(sy_c_ATP_058Lamm__ago____,type,
    aTP_Lamm_ago: set_nat > fun_fu4637029388598527722ex_nat ).

tff(sy_c_ATP_058Lamm__agp____,type,
    aTP_Lamm_agp: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__agq____,type,
    aTP_Lamm_agq: set_complex > fun_fu1598310740840436355x_bool ).

tff(sy_c_ATP_058Lamm__agr____,type,
    aTP_Lamm_agr: set_complex > fun_fu1817986637694864396at_nat ).

tff(sy_c_ATP_058Lamm__ags____,type,
    aTP_Lamm_ags: set_nat > fun_fu1093554848263573315t_bool ).

tff(sy_c_ATP_058Lamm__agt____,type,
    aTP_Lamm_agt: set_nat > fun_fu1709340832601392443er_nat ).

tff(sy_c_ATP_058Lamm__agu____,type,
    aTP_Lamm_agu: fun_Co3312552843764780407at_nat > fun_na5492297256294505719er_nat ).

tff(sy_c_ATP_058Lamm__agv____,type,
    aTP_Lamm_agv: set_Code_integer > fun_fu4694495798485950403r_bool ).

tff(sy_c_ATP_058Lamm__agw____,type,
    aTP_Lamm_agw: set_Code_integer > fun_fu5158082794216327598at_nat ).

tff(sy_c_ATP_058Lamm__agx____,type,
    aTP_Lamm_agx: set_complex > fun_fu5159174019657214751x_bool ).

tff(sy_c_ATP_058Lamm__agy____,type,
    aTP_Lamm_agy: set_complex > fun_fu8973321760839499616omplex ).

tff(sy_c_ATP_058Lamm__agz____,type,
    aTP_Lamm_agz: fun_VE2006399705683203104omplex > fun_co3541073130412562976omplex ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__aha____,type,
    aTP_Lamm_aha: set_VEBT_VEBT > fun_fu3138954370166276442omplex ).

tff(sy_c_ATP_058Lamm__ahb____,type,
    aTP_Lamm_ahb: set_complex > fun_fu4189668387880940419x_bool ).

tff(sy_c_ATP_058Lamm__ahc____,type,
    aTP_Lamm_ahc: set_complex > fun_fu3241646296630235056omplex ).

tff(sy_c_ATP_058Lamm__ahd____,type,
    aTP_Lamm_ahd: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__ahe____,type,
    aTP_Lamm_ahe: set_real > fun_fu7272995576155203506omplex ).

tff(sy_c_ATP_058Lamm__ahf____,type,
    aTP_Lamm_ahf: set_complex > fun_fu8281188428872102787x_bool ).

tff(sy_c_ATP_058Lamm__ahg____,type,
    aTP_Lamm_ahg: set_complex > fun_fu5276431085538847372omplex ).

tff(sy_c_ATP_058Lamm__ahh____,type,
    aTP_Lamm_ahh: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ahi____,type,
    aTP_Lamm_ahi: set_nat > fun_fu2548079958729413763t_bool ).

tff(sy_c_ATP_058Lamm__ahj____,type,
    aTP_Lamm_ahj: set_nat > fun_fu5286008751397070186omplex ).

tff(sy_c_ATP_058Lamm__ahk____,type,
    aTP_Lamm_ahk: set_complex > fun_fu3391973010043818627x_bool ).

tff(sy_c_ATP_058Lamm__ahl____,type,
    aTP_Lamm_ahl: set_complex > fun_fu2147946303232500784omplex ).

tff(sy_c_ATP_058Lamm__ahm____,type,
    aTP_Lamm_ahm: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__ahn____,type,
    aTP_Lamm_ahn: set_int > fun_fu8949065344085950851t_bool ).

tff(sy_c_ATP_058Lamm__aho____,type,
    aTP_Lamm_aho: set_int > fun_fu2298032621795249330omplex ).

tff(sy_c_ATP_058Lamm__ahp____,type,
    aTP_Lamm_ahp: set_complex > fun_fu133147570795794627x_bool ).

tff(sy_c_ATP_058Lamm__ahq____,type,
    aTP_Lamm_ahq: set_complex > fun_fu2933621645842287103omplex ).

tff(sy_c_ATP_058Lamm__ahr____,type,
    aTP_Lamm_ahr: fun_Co8905303033605938227omplex > fun_co1305099417915878579omplex ).

tff(sy_c_ATP_058Lamm__ahs____,type,
    aTP_Lamm_ahs: set_Code_integer > fun_fu4200121666219089987r_bool ).

tff(sy_c_ATP_058Lamm__aht____,type,
    aTP_Lamm_aht: set_Code_integer > fun_fu3955559650531061584omplex ).

tff(sy_c_ATP_058Lamm__ahu____,type,
    aTP_Lamm_ahu: fun_VE5569070535936029429omplex > fun_fu6701721618212012456omplex ).

tff(sy_c_ATP_058Lamm__ahv____,type,
    aTP_Lamm_ahv: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__ahw____,type,
    aTP_Lamm_ahw: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__ahx____,type,
    aTP_Lamm_ahx: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__ahy____,type,
    aTP_Lamm_ahy: set_Code_integer > fun_fu9052069482527730051r_bool ).

tff(sy_c_ATP_058Lamm__ahz____,type,
    aTP_Lamm_ahz: fun_Co1900716531650368264omplex > fun_fu6624223949315877998omplex ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_real_real ).

tff(sy_c_ATP_058Lamm__aia____,type,
    aTP_Lamm_aia: fun_VEBT_VEBT_real > fun_fu5928985204125839270T_real ).

tff(sy_c_ATP_058Lamm__aib____,type,
    aTP_Lamm_aib: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__aic____,type,
    aTP_Lamm_aic: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__aid____,type,
    aTP_Lamm_aid: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__aie____,type,
    aTP_Lamm_aie: fun_Co3960896333048504454r_real > fun_fu6908834162160313196r_real ).

tff(sy_c_ATP_058Lamm__aif____,type,
    aTP_Lamm_aif: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aig____,type,
    aTP_Lamm_aig: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__aih____,type,
    aTP_Lamm_aih: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__aii____,type,
    aTP_Lamm_aii: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aij____,type,
    aTP_Lamm_aij: fun_nat_real ).

tff(sy_c_ATP_058Lamm__aik____,type,
    aTP_Lamm_aik: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ail____,type,
    aTP_Lamm_ail: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aim____,type,
    aTP_Lamm_aim: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ain____,type,
    aTP_Lamm_ain: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aio____,type,
    aTP_Lamm_aio: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aip____,type,
    aTP_Lamm_aip: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aiq____,type,
    aTP_Lamm_aiq: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__air____,type,
    aTP_Lamm_air: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ais____,type,
    aTP_Lamm_ais: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ait____,type,
    aTP_Lamm_ait: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aiu____,type,
    aTP_Lamm_aiu: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aiv____,type,
    aTP_Lamm_aiv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aiw____,type,
    aTP_Lamm_aiw: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__aix____,type,
    aTP_Lamm_aix: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__aiy____,type,
    aTP_Lamm_aiy: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aiz____,type,
    aTP_Lamm_aiz: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__aja____,type,
    aTP_Lamm_aja: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajb____,type,
    aTP_Lamm_ajb: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ajc____,type,
    aTP_Lamm_ajc: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ajd____,type,
    aTP_Lamm_ajd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aje____,type,
    aTP_Lamm_aje: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajf____,type,
    aTP_Lamm_ajf: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ajg____,type,
    aTP_Lamm_ajg: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajh____,type,
    aTP_Lamm_ajh: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aji____,type,
    aTP_Lamm_aji: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajj____,type,
    aTP_Lamm_ajj: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ajk____,type,
    aTP_Lamm_ajk: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajl____,type,
    aTP_Lamm_ajl: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajm____,type,
    aTP_Lamm_ajm: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajn____,type,
    aTP_Lamm_ajn: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ajo____,type,
    aTP_Lamm_ajo: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ajp____,type,
    aTP_Lamm_ajp: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ajq____,type,
    aTP_Lamm_ajq: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ajr____,type,
    aTP_Lamm_ajr: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ajs____,type,
    aTP_Lamm_ajs: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ajt____,type,
    aTP_Lamm_ajt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aju____,type,
    aTP_Lamm_aju: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ajv____,type,
    aTP_Lamm_ajv: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ajw____,type,
    aTP_Lamm_ajw: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ajx____,type,
    aTP_Lamm_ajx: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ajy____,type,
    aTP_Lamm_ajy: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ajz____,type,
    aTP_Lamm_ajz: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aka____,type,
    aTP_Lamm_aka: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__akb____,type,
    aTP_Lamm_akb: fun_real_bool ).

tff(sy_c_ATP_058Lamm__akc____,type,
    aTP_Lamm_akc: fun_re8661203372455305201t_bool ).

tff(sy_c_ATP_058Lamm__akd____,type,
    aTP_Lamm_akd: rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ake____,type,
    aTP_Lamm_ake: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__akf____,type,
    aTP_Lamm_akf: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__akg____,type,
    aTP_Lamm_akg: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__akh____,type,
    aTP_Lamm_akh: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__aki____,type,
    aTP_Lamm_aki: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__akj____,type,
    aTP_Lamm_akj: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__akk____,type,
    aTP_Lamm_akk: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__akl____,type,
    aTP_Lamm_akl: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__akm____,type,
    aTP_Lamm_akm: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__akn____,type,
    aTP_Lamm_akn: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__ako____,type,
    aTP_Lamm_ako: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__akp____,type,
    aTP_Lamm_akp: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__akq____,type,
    aTP_Lamm_akq: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__akr____,type,
    aTP_Lamm_akr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aks____,type,
    aTP_Lamm_aks: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__akt____,type,
    aTP_Lamm_akt: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aku____,type,
    aTP_Lamm_aku: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__akv____,type,
    aTP_Lamm_akv: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__akw____,type,
    aTP_Lamm_akw: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__akx____,type,
    aTP_Lamm_akx: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__aky____,type,
    aTP_Lamm_aky: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__akz____,type,
    aTP_Lamm_akz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_int_int ).

tff(sy_c_ATP_058Lamm__ala____,type,
    aTP_Lamm_ala: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__alb____,type,
    aTP_Lamm_alb: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__alc____,type,
    aTP_Lamm_alc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ald____,type,
    aTP_Lamm_ald: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ale____,type,
    aTP_Lamm_ale: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__alf____,type,
    aTP_Lamm_alf: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__alg____,type,
    aTP_Lamm_alg: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__alh____,type,
    aTP_Lamm_alh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ali____,type,
    aTP_Lamm_ali: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__alj____,type,
    aTP_Lamm_alj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__alk____,type,
    aTP_Lamm_alk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__all____,type,
    aTP_Lamm_all: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__alm____,type,
    aTP_Lamm_alm: nat > fun_na5042103099923536755t_bool ).

tff(sy_c_ATP_058Lamm__aln____,type,
    aTP_Lamm_aln: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__alo____,type,
    aTP_Lamm_alo: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__alp____,type,
    aTP_Lamm_alp: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__alq____,type,
    aTP_Lamm_alq: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__alr____,type,
    aTP_Lamm_alr: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__als____,type,
    aTP_Lamm_als: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: set_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: vEBT_VEBT > fun_VE4942641508680916543l_assn ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

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

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_real_bool ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_bool_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_int_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

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

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_VEBT_VEBT > fun_fu5250730037759307465T_bool ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_list_nat_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_list_nat_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_set_nat_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_set_nat_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_nat_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_nat_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_int_nat > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_complex_nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_complex_nat > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_Code_integer_nat > fun_Co8822979462182099027r_bool ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_Code_integer_nat > fun_fu9052069482527730051r_bool ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_list_nat_nat > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_set_nat_nat > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_int_nat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_complex_nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_Code_integer_nat > fun_Co8310303586393106575r_bool ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_complex > fun_set_complex_bool ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: set_Code_integer > fun_se2712834123562403997r_bool ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_se6049520964208396421t_bool ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_Co8822979462182099027r_bool > fun_fu9052069482527730051r_bool ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_Co8822979462182099027r_bool > fun_fu9052069482527730051r_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_VEBT_VEBT > fun_fu2721280555464079775T_bool ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_real > fun_fu4470565429337120259l_bool ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_VEBT_VEBT > fun_fu7178057608955062431T_bool ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_real > fun_fu66453571216240387l_bool ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_VEBT_VEBT > fun_fu748158482767045919T_bool ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_real > fun_fu6440467031773438595l_bool ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_VEBT_VEBT > fun_fu7855050327546112735T_bool ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_real > fun_fu6700586743015562435l_bool ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_nat > fun_fu4970875099359191043t_bool ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_nat > fun_fu7165551060251750659t_bool ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_real_real ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_int_int ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_assn_assn ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_se2759736925585322443t_bool ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_se1152882363804747413t_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_se1071123033157107803t_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_se2083251475300338071r_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_se3316967244840431159m_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_Code_integer > fun_fu355156309537052966r_bool ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_Code_integer > fun_fu347749604005694585r_bool ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_VEBT_VEBT > fun_fu3463172280133395800T_bool ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_VEBT_VEBT > fun_fu1533684823456897980T_bool ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_real > fun_fu2277326744902863452l_bool ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_real > fun_fu6750725090257165812l_bool ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_nat > fun_fu3052598110916186900t_bool ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_nat > fun_fu2971308621445215880t_bool ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_int > fun_fu337402001079966300t_bool ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_int > fun_fu163127709195666036t_bool ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: vEBT_VEBT > fun_VE3808733824349961398l_assn ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_VEBT_VEBT > fun_fu8277615433164260398T_bool ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_VEBT_VEBT > fun_fu1247336914219032636T_bool ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: set_Code_integer > fun_fu8681741693558125220r_bool ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_Code_integer > fun_fu3125884428166449269r_bool ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: set_Code_integer > fun_na3912021598713191116r_bool ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_real > fun_na6137814669449589053l_bool ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: set_Code_integer > fun_na3912021598713191116r_bool ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: set_real > fun_na6137814669449589053l_bool ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: nat > fun_li5407204298491981421T_bool ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: nat > fun_li4271802897377934289l_bool ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: nat > fun_li402716648095076269l_bool ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: nat > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: nat > fun_li582272206764200913t_bool ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_bool_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: list_VEBT_VEBTi > fun_na7681084296160214226_VEBTi ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: list_VEBT_VEBT > fun_na2183005016389635715T_VEBT ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: list_P6011104703257516679at_nat > fun_na7803622437922444752at_nat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: list_num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: list_real > fun_nat_option_real ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: list_bool > fun_nat_option_bool ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: list_nat > fun_nat_option_nat ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: list_int > fun_nat_option_int ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_VEBT_VEBT_real > fun_VE5877677516038742509T_bool ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_nat_rat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_int_rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_nat_num > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_int_num > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_VEBT_VEBT_nat > fun_VE5877677516038742509T_bool ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_int_nat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_nat_int > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_re1255401033344146673l_bool > fun_fu982491298992145277T_bool ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_bo1841576699660343469l_bool > fun_fu1522570694743846419l_bool ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_bo1841576699660343469l_bool > fun_fu4978346893710585036t_bool ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_fun_nat_bool > fun_fu7019631248157564889T_bool ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_nat_fun_nat_bool > fun_fu3912789684692335013t_bool ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_nat_fun_nat_bool > fun_fu2873736630187781521t_bool ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_int_fun_int_bool > fun_fu2740518503626799177t_bool ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_int_fun_int_bool > fun_fu1701465449122245685t_bool ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_list_nat_nat > fun_fu2359051559149003772t_bool ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_set_nat_nat > fun_fu8021297703010766256t_bool ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_nat > fun_fu8422113094057456412t_bool ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_int_nat > fun_fu5706916984221235812t_bool ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_complex_nat > fun_fu5450758583235971936x_bool ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_Code_integer_nat > fun_fu7000754783045315842r_bool ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: assn > fun_fu8734043002210220995t_assn ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: assn > fun_fu8831737957336725536t_assn ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: assn > fun_fu2283375113763962243t_assn ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: assn > fun_fu575597352165939680t_assn ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: assn > fun_fu2678428474534703935l_assn ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: assn > fun_fu1723287682096855374l_assn ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: assn > fun_fu5995910667641283065l_assn ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: assn > fun_fu3058217449524045576l_assn ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: assn > fun_fu2268873205120543299i_assn ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: assn > fun_fu7814006171152732576i_assn ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: assn > fun_fu8979060164380311043i_assn ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: assn > fun_fu8850137702660867424i_assn ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: assn > fun_fu96848488177702211t_assn ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: assn > fun_fu1092876921774764064t_assn ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: assn > fun_fu1565617872119973507t_assn ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: assn > fun_fu8295659832165499232t_assn ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: nat > fun_VEBT_VEBT_nat ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: real > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: vEBT_VEBT > fun_real_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: vEBT_VEBT > fun_bool_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_bo1841576699660343469l_bool ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_bo3288154511880838028t_bool ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: vEBT_VEBT > fun_int_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: set_int > fun_se6049520964208396421t_bool ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: set_complex > fun_se2925769915755310977x_bool ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: set_Code_integer > fun_se4340818509347730467r_bool ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_nat > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: list_VEBT_VEBT > fun_VE935608329480070972T_VEBT ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: list_real > fun_VE6599034769933093990T_real ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: list_bool > fun_VE8273198136501265567T_bool ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: list_VEBT_VEBT > fun_na2031173642954462902T_VEBT ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: list_real > fun_na1129888583756534636t_real ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: list_bool > fun_na7570524936995945497t_bool ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: list_VEBT_VEBT > fun_re7324851661301934134T_VEBT ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: list_nat > fun_re2265854798118090896al_nat ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: list_real > fun_re8903365113552904684l_real ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: list_bool > fun_re2935828335731428761l_bool ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_VE5877677516038742509T_bool ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_bo1841576699660343469l_bool ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: set_VEBT_VEBT > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: set_complex > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: set_list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: set_set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: vEBT_VEBT > fun_fu5250730037759307465T_bool ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: bool > fun_fu7925023953348749699l_bool ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: list_nat > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: set_nat > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: bool > fun_se689384475314470797l_bool ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: vEBT_VEBT > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: complex > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_VEBT_VEBT_bool > fun_VE5877677516038742509T_bool ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_complex_bool > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_bool_bool > fun_bo1841576699660343469l_bool ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_VEBT_VEBT_bool > fun_VE5877677516038742509T_bool ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_complex_bool > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_list_nat_bool > fun_li8598321315282354825t_bool ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_set_nat_bool > fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_bool_bool > fun_bo1841576699660343469l_bool ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_int_bool > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: list_nat > fun_na1977831282410318352at_nat ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: list_nat > fun_VE4168426245860933642BT_nat ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: list_int > fun_in3390538432181454188nt_int ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: list_Code_integer > fun_Co998730852857036893nteger ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: list_num > fun_na8171767993680280922at_num ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: set_bool > fun_fu8231870962471999726l_bool ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: set_Code_integer > fun_fu355156309537052966r_bool ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: set_bool > fun_fu2238024911088387052l_bool ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: set_bool > fun_fu2238024911088387052l_bool ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: set_bool > fun_fu6678276313180071960l_bool ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_VE5569070535936029429omplex > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_real_complex > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_complex_complex > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_nat_complex > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_int_complex > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_VEBT_VEBT_real > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_real_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_complex_real > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_nat_real > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_int_real > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_VEBT_VEBT_assn > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_real_assn > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_complex_assn > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_assn > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_int_assn > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_VEBT_VEBT_real > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_real_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_complex_real > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_real > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_int_real > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_Co3960896333048504454r_real > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_Code_integer_rat > fun_fu722387130092441185er_rat ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_Co3960896333048504454r_real > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_Code_integer_rat > fun_fu722387130092441185er_rat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_VE5569070535936029429omplex > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_real_complex > fun_fu8384527901749275179omplex ).

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

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_int_complex > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_Co1900716531650368264omplex > fun_fu3438678930028006669omplex ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_VEBT_VEBT_real > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_VEBT_VEBT_real > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_VEBT_VEBT_rat > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_VE6757229793580780342T_VEBT ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_re5118428557197245286l_real ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_bo8995055541810095699l_bool ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_in4211310926729683942nt_int ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_Co6597186368970283863nteger ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: set_Code_integer > fun_na3912021598713191116r_bool ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: set_real > fun_na6137814669449589053l_bool ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: set_VEBT_VEBT > fun_li6006399982103069984T_bool ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: set_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: set_real > fun_list_real_bool ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: set_bool > fun_list_bool_bool ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: set_VEBT_VEBT > fun_li6006399982103069984T_bool ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: set_real > fun_list_real_bool ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: set_bool > fun_list_bool_bool ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: set_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: set_complex > fun_li356976171784545970x_bool ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: set_Code_integer > fun_li7102945298704276035r_bool ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: set_int > fun_list_int_bool ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_VE1225457145571040039t_assn > fun_li8822687668615115362l_assn ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_VE2561185598974572172i_assn > fun_li8369987531566349255l_assn ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_VE5577048379444288083T_assn > fun_li1808587207459806180l_assn ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_VE5641251438918192003t_assn > fun_li4226673942822278846l_assn ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_VE4931415397359990460t_assn > fun_li7943157569623896045l_assn ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_VE1571388752190420990T_assn > fun_li2158447775985626393l_assn ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_VE123837653852366616t_assn > fun_li3347143843831059529l_assn ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_re574865750389657758t_assn > fun_li204261821335639001l_assn ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_re560840401387337987i_assn > fun_li2608363201533078078l_assn ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_re8050108611700907676T_assn > fun_li7232704599769625645l_assn ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_li51272947660841601i_assn ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: nat > fun_li8672845860709154922t_assn ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: nat > fun_li7646410839599160030t_assn ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: nat > fun_li8221786654549472865l_assn ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: nat > fun_li6689927339917241139i_assn ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: nat > fun_li6911922195568332558t_assn ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: array_VEBT_VEBTi > fun_li7822782016457402668i_assn ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: nat > fun_li5996000409811367950l_assn ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: nat > fun_li714799757257233060t_assn ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: nat > fun_li92960653718430884t_assn ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: nat > fun_li3423855071176226084i_assn ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_VE2561185598974572172i_assn > fun_li7603156796965727646n_assn ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_VE5577048379444288083T_assn > fun_li3408021040229202491n_assn ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_VE1571388752190420990T_assn > fun_li8179631313216443376n_assn ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_re560840401387337987i_assn > fun_li1595705456081383829n_assn ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_re8050108611700907676T_assn > fun_li5438311264709876996n_assn ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_bo7535889243153246212i_assn > fun_li968352931858857356n_assn ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_bo5394993412183903963T_assn > fun_li3553553328591960013n_assn ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_na6467670216208870567i_assn > fun_li8632452286561901497n_assn ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_na4489259242115493880T_assn > fun_li5978570208557324256n_assn ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_in1903134756389476099i_assn > fun_li135755063789416981n_assn ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: array_bool > fun_li6008471108080514341t_assn ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: array_nat > fun_li4928566814967809234t_assn ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: array_int > fun_li1703715468755832110t_assn ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: array_VEBT_VEBTi > fun_li7023630697154767927t_assn ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_real_real > fun_re3005535450163344039l_real ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_bool_int > fun_in3606259060872855084nt_int ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_VEBT_VEBT_nat > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: nat > fun_bo7626686174254799871l_assn ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: nat > fun_VE863238786339739537i_assn ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_VE2561185598974572172i_assn > fun_VE5858173938467412610n_assn ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_VE5577048379444288083T_assn > fun_VE7753015393293352042n_assn ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_VE905575822219544865i_assn > fun_VE5501718045660582762n_assn ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_VE905575822219544865i_assn > fun_li2168009211280042921n_assn ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_VE1571388752190420990T_assn > fun_VE9159209985096290642n_assn ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_VE1225457145571040039t_assn > fun_VE6630131001635098680n_assn ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_VE1225457145571040039t_assn > fun_li9171513756222907321n_assn ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_VE4931415397359990460t_assn > fun_VE8014634195917101344n_assn ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_VE4931415397359990460t_assn > fun_li3142387651386734788n_assn ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_VE5641251438918192003t_assn > fun_VE3362914947041059568n_assn ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_VE5641251438918192003t_assn > fun_li4974362004767202837n_assn ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_VE123837653852366616t_assn > fun_VE4747418141323062232n_assn ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_VE123837653852366616t_assn > fun_li8168607936785806112n_assn ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_re560840401387337987i_assn > fun_re5421100873081382000n_assn ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_re8050108611700907676T_assn > fun_re6966045276204452696n_assn ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_nat_fun_bool_nat ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: list_list_nat > fun_na8717435504434197565st_nat ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: list_list_int > fun_in1228059937145375477st_int ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: list_nat > fun_nat_list_nat ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: list_nat > fun_li8329949339774906573st_nat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: list_int > fun_int_list_int ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: list_int > fun_li4724518053915367813st_int ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_co2535609921677657902omplex ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_na8167864584424996432at_nat ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_in1793339680377598124nt_int ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_Co4731079614609534941nteger ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_real_real ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_int_int ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_VEBT_VEBT_bool > fun_VE8973593669298627338T_VEBT ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_complex_bool > fun_co8804923750838637808omplex ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_Co8822979462182099027r_bool > fun_Co1555053522382583933nteger ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_real_bool > fun_re5446434971879228394t_real ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_bool_bool > fun_bo2177052459101130925t_bool ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_bool > fun_na7754705478876614998et_nat ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_int_bool > fun_in3946961816151909994et_int ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: set_VEBT_VEBT > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: set_real > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: set_complex > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: set_list_nat > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: set_set_nat > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: set_int > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: set_nat > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_VEBT_VEBT_bool > fun_se6960202813843830349T_bool ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_real_bool > fun_se2197967325210717499l_bool ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_complex_bool > fun_se2973560062716774839x_bool ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_list_nat_bool > fun_se7035039998181355731t_bool ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_set_nat_bool > fun_se1925730536474940935t_bool ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_nat_bool > fun_se680526843285281907t_bool ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_int_bool > fun_se6612071603986899387t_bool ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_as7903617535776176121n_bool ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_as7903617535776176121n_bool ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_se1377969734402345349l_bool ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_se1377969734402345349l_bool ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_se7834784120450421933l_bool ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_se7834784120450421933l_bool ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_se707272970763642173t_bool ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_se6049520964208396421t_bool ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_bool_bool ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_num_num ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_as3983628580203722482n_assn ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_real_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_real_nat > fun_fu1614898059344306963at_nat ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_real_int > fun_fu8348186951267356015at_int ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_int_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_int_nat > fun_fu1731011121079923219at_nat ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_nat_int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_nat_int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_int_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_VE3165209603783583043T_VEBT > fun_se8982498966842174448T_bool ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_VE3165209603783583043T_VEBT > fun_se8982498966842174448T_bool ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_real_VEBT_VEBT > fun_se800027511083410856T_bool ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_real_VEBT_VEBT > fun_se3953343959147634926l_bool ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_VEBT_VEBT_real > fun_se3531477064570318058l_bool ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_VEBT_VEBT_real > fun_se2555844317841001132T_bool ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_real_real > fun_se3079162737911483042l_bool ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_real_real > fun_se3079162737911483042l_bool ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_co4340422964966740447T_VEBT > fun_se5627406473294553258T_bool ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_co4340422964966740447T_VEBT > fun_se2530540278237434730x_bool ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_complex_real > fun_se4123101102803514532l_bool ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_complex_real > fun_se6962783776300733858x_bool ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_nat_VEBT_VEBT > fun_se1104498734146315468T_bool ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_VEBT_VEBT > fun_se939662418109020198t_bool ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_nat_real > fun_se4590301831022467014l_bool ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_nat_real > fun_se4876108031464829730t_bool ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_int_VEBT_VEBT > fun_se2716886841848879400T_bool ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_int_VEBT_VEBT > fun_se6022440845063412078t_bool ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_int_real > fun_se3247505022010441250l_bool ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_int_real > fun_se3752721087501804578t_bool ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: set_VEBT_VEBT > fun_fu7019631248157564889T_bool ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: set_real > fun_fu707799363171429393l_bool ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: set_VEBT_VEBT > fun_fu3026365786366995581T_bool ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: set_real > fun_fu4724935602937317045l_bool ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: set_VEBT_VEBT > fun_fu4038965111066273403T_bool ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: set_real > fun_fu5560604037825259955l_bool ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: set_VEBT_VEBT > fun_fu6282964682757597068T_bool ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: set_real > fun_fu2400374679489434180l_bool ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: set_nat > fun_fu9204309250095613769t_bool ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: set_nat > fun_fu3912789684692335013t_bool ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: vEBT_VEBT > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: real > fun_real_real ).

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

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: bool > fun_VEBT_VEBT_bool ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: nat > fun_real_nat ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: int > fun_real_int ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: real > fun_bool_real ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: nat > fun_bool_nat ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: int > fun_bool_int ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: nat > fun_VE3025544856504383354BT_nat ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: code_integer > fun_Co6597186368970283863nteger ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: vEBT_VEBT > fun_re2335940313395306534T_VEBT ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: bool > fun_re8087791432033743625l_bool ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: vEBT_VEBT > fun_bo900138793126669168T_VEBT ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: bool > fun_bo8995055541810095699l_bool ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: vEBT_VEBT > fun_na2623420418515794982T_VEBT ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: bool > fun_na3550481584588361481t_bool ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: nat > fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: num > fun_na6415222038798377684at_num ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_VEBT_VEBT > fun_na5194437689033895312T_VEBT ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_int_VEBT_VEBT > fun_in5638885446023802348T_VEBT ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_co4340422964966740447T_VEBT > fun_co838889460547638638T_VEBT ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_Co3766629010929971184T_VEBT > fun_Co3368914069033135517T_VEBT ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_real > fun_na7753251452534370190t_real ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_int_real > fun_in6227199644792676842t_real ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_complex_real > fun_co7713368719948855404t_real ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_Co3960896333048504454r_real > fun_Co283803684186153243t_real ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_nat_bool > fun_na1053657639205449936t_bool ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_int_bool > fun_in8682626120193986860t_bool ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_rat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_complex_bool > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_list_nat_bool > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_set_nat_bool > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_int_bool > fun_int_bool ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: set_VEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: set_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: set_list_nat > fun_list_nat_bool ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: set_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: fun_complex_bool > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: fun_list_nat_bool > fun_fu8303687057036190147t_bool ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_set_nat_bool > fun_fu7495562154832025027t_bool ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_int_bool > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: fun_nat_bool > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: int > fun_in4211310926729683942nt_int ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: nat > fun_na7044673412300895210at_nat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: nat > fun_na8167864584424996432at_nat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: vEBT_VEBT > fun_na6995080267563725110BT_nat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: vEBT_VEBT > fun_na3147228585898040220BT_nat ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: int > fun_in7209865251430457670nt_int ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: int > fun_in1793339680377598124nt_int ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: code_integer > fun_Co2443134212982465463nteger ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: code_integer > fun_Co4731079614609534941nteger ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: nat > fun_nu5900142480299520106at_num ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: nat > fun_nu8275335919628737562at_num ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: nat > fun_int_nat ).

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

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

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: int > fun_complex_int ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: int > fun_Code_integer_int ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_real_set_nat > fun_re7314947225183814911et_nat ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_bool_set_nat > fun_bo6580201250007910272et_nat ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: fun_nat_set_nat > fun_na896109329399166627et_nat ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_int_set_nat > fun_in2104226462327546623et_nat ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: list_set_real > fun_set_real_bool ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: list_set_bool > fun_set_bool_bool ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: list_set_int > fun_set_int_bool ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: list_set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: fun_complex_real > fun_re2354894731504631552x_real ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: fun_Co3960896333048504454r_real > fun_re7921287988085964207r_real ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_complex_rat > fun_ra2209366104776213208ex_rat ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: fun_Code_integer_rat > fun_ra3321464020636160263er_rat ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: fun_real_rat > fun_rat_fun_real_rat ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: fun_bool_real > fun_re7210842476993315843l_real ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_bool_rat > fun_rat_fun_bool_rat ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: fun_na7044673412300895210at_nat ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_in7209865251430457670nt_int ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_Co2443134212982465463nteger ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_real_set_real > fun_se5043357129389602794t_real ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_real_set_bool > fun_se6830452941867646896t_bool ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_real_set_int > fun_se3478072698188602730et_int ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_bool_set_real > fun_se5411068930143410819t_real ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_bool_set_bool > fun_se7698381193922241133t_bool ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_bool_set_int > fun_se6410524667074557315et_int ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_nat_set_real > fun_se3856533179008017806t_real ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_nat_set_bool > fun_se6063655520620159884t_bool ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_nat_set_int > fun_se2974063760341720334et_int ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_int_set_real > fun_se5704610510406758378t_real ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: set_real > fun_fu4680957323884239763t_real ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: set_bool > fun_fu3821001101017267907t_bool ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: set_int > fun_fu4579819628630762643et_int ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: set_real > fun_fu2645947647283103939t_real ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: set_bool > fun_fu7192499100992716611t_bool ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: set_int > fun_fu8556761082339699011et_int ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: set_real > fun_fu1480296319422391259t_real ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: set_bool > fun_fu1091034382302749763t_bool ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: set_int > fun_fu4581781630362247899et_int ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: set_real > fun_fu779935128596377747t_real ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: vEBT_VEBT > fun_fu8357240957987687427T_VEBT ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: real > fun_fu4680957323884239763t_real ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: bool > fun_fu3821001101017267907t_bool ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: int > fun_fu4579819628630762643et_int ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: vEBT_VEBT > fun_fu8893799565311400573T_VEBT ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: real > fun_fu2645947647283103939t_real ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: bool > fun_fu7192499100992716611t_bool ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: int > fun_fu8556761082339699011et_int ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: vEBT_VEBT > fun_fu598773264467018499T_VEBT ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: real > fun_fu1480296319422391259t_real ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_VE3640887651371881891T_VEBT ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_real_set_real ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_bool_set_bool ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_int_set_int ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_set_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: set_nat > fun_real_set_nat ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: set_nat > fun_bool_set_nat ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: set_nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: set_nat > fun_int_set_nat ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: set_real > fun_real_set_real ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: set_bool > fun_real_set_bool ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: set_int > fun_real_set_int ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: set_real > fun_bool_set_real ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: set_bool > fun_bool_set_bool ).

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_Array__Time_Ofreeze_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_8141364883501958055_VEBTi: array_VEBT_VEBTi > heap_T4980287057938770641_VEBTi ).

tff(sy_c_Array__Time_Olen_001t__HOL__Obool,type,
    array_len_bool: array_bool > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Olen_001t__Int__Oint,type,
    array_len_int: array_int > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Olen_001t__Nat__Onat,type,
    array_len_nat: array_nat > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Olen_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_len_VEBT_VEBTi: array_VEBT_VEBTi > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Onew_001t__HOL__Obool,type,
    array_new_bool: ( nat * bool ) > heap_T6374562886178639292y_bool ).

tff(sy_c_Array__Time_Onew_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_new_VEBT_VEBTi: ( nat * vEBT_VEBTi ) > heap_T8822477325091257596_VEBTi ).

tff(sy_c_Array__Time_Onth_001t__HOL__Obool,type,
    array_nth_bool: ( array_bool * nat ) > heap_Time_Heap_bool ).

tff(sy_c_Array__Time_Onth_001t__Int__Oint,type,
    array_nth_int: ( array_int * nat ) > heap_Time_Heap_int ).

tff(sy_c_Array__Time_Onth_001t__Nat__Onat,type,
    array_nth_nat: ( array_nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    array_nth_option_nat: ( array_option_nat * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_Array__Time_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_nth_VEBT_VEBTi: ( array_VEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_Array__Time_Oof__list_001t__HOL__Obool,type,
    array_of_list_bool: list_bool > heap_T6374562886178639292y_bool ).

tff(sy_c_Array__Time_Oof__list_001t__Int__Oint,type,
    array_of_list_int: list_int > heap_T1346037964561226099ay_int ).

tff(sy_c_Array__Time_Oof__list_001t__Nat__Onat,type,
    array_of_list_nat: list_nat > heap_T3836121109492952855ay_nat ).

tff(sy_c_Array__Time_Oof__list_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_615059503499738533_VEBTi: list_VEBT_VEBTi > heap_T8822477325091257596_VEBTi ).

tff(sy_c_Array__Time_Oupd_001t__HOL__Obool,type,
    array_upd_bool: ( nat * bool * array_bool ) > heap_T6374562886178639292y_bool ).

tff(sy_c_Array__Time_Oupd_001t__Int__Oint,type,
    array_upd_int: ( nat * int * array_int ) > heap_T1346037964561226099ay_int ).

tff(sy_c_Array__Time_Oupd_001t__Nat__Onat,type,
    array_upd_nat: ( nat * nat * array_nat ) > heap_T3836121109492952855ay_nat ).

tff(sy_c_Array__Time_Oupd_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_upd_VEBT_VEBTi: ( nat * vEBT_VEBTi * array_VEBT_VEBTi ) > heap_T8822477325091257596_VEBTi ).

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

tff(sy_c_Assertions_Oprecise_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    precis1518117471652397933_VEBTi: fun_li51272947660841601i_assn > $o ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: fun_bool_assn ).

tff(sy_c_Assertions_Osnga__assn_001t__HOL__Obool,type,
    snga_assn_bool: ( array_bool * list_bool ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Int__Oint,type,
    snga_assn_int: ( array_int * list_int ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Nat__Onat,type,
    snga_assn_nat: ( array_nat * list_nat ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Option__Ooption_It__Nat__Onat_J,type,
    snga_assn_option_nat: ( array_option_nat * list_option_nat ) > 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_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr3099820731930151597t_bool ).

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

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

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: fun_int_fun_nat_bool ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_bool ).

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

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
    bit_Sh3965577149348748681tl_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr_001t__Nat__Onat,type,
    bit_Sh2154871086232339855tr_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: fun_set_nat_nat ).

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

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__HOL__Obool,type,
    comple6445916836919887748p_bool: fun_set_bool_bool ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
    complete_Sup_Sup_int: fun_set_int_int ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: fun_set_nat_nat ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Option__Ooption_It__HOL__Obool_J,type,
    comple4504012738199311306n_bool: set_option_bool > option_bool ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    comple8455683388168444585et_nat: set_option_set_nat > option_set_nat ).

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

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    comple739944243200306918nteger: set_set_Code_integer > set_Code_integer ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    comple8424636186594484919omplex: set_set_complex > set_complex ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__HOL__Obool_J,type,
    comple9167723193526752868t_bool: set_set_bool > set_bool ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_OCons_001t__Rat__Orat,type,
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tff(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_OCons_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Olist_OCons_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_ONil_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_ONil_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__HOL__Obool_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_List_Olist_Omap_001t__List__Olist_It__HOL__Obool_J_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__List__Olist_It__Int__Oint_J_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__List__Olist_It__Real__Oreal_J_001t__Nat__Onat,type,
    map_list_real_nat: ( fun_list_real_nat * list_list_real ) > list_nat ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__HOL__Obool_J,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Rat__Orat,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Nat__Onat_J,type,
    map_VE3614494202555519440st_nat: ( fun_VE4564399238913696295st_nat * list_VEBT_VEBT ) > list_list_nat ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J_J,type,
    map_VE8357768277379645088T_bool: ( fun_VE8273198136501265567T_bool * list_VEBT_VEBT ) > list_l5795148099979146390T_bool ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    map_VE4456067000678548787BT_nat: ( fun_VE4168426245860933642BT_nat * list_VEBT_VEBT ) > list_l6108278773223375443BT_nat ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
    map_VE6455195641817765263T_real: ( fun_VE6599034769933093990T_real * list_VEBT_VEBT ) > list_l1703466484006042159T_real ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    map_VE6710161768887062461T_VEBT: ( fun_VE935608329480070972T_VEBT * list_VEBT_VEBT ) > list_l9025237890452396851T_VEBT ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Real__Oreal_J,type,
    map_VE755361874264402988t_real: ( fun_VE8613710830879766659t_real * list_VEBT_VEBT ) > list_list_real ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    map_VE5619196432070217417T_VEBT: ( fun_VE1635941959115973792T_VEBT * list_VEBT_VEBT ) > list_P5647936690300460905T_VEBT ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    map_VE1210512188453230445T_VEBT: ( fun_VE3519909879894771012T_VEBT * list_VEBT_VEBT ) > list_P877281246627933069T_VEBT ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    map_VE6666121349297532579BT_nat: ( fun_VE3025544856504383354BT_nat * list_VEBT_VEBT ) > list_P7037539587688870467BT_nat ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    map_VE1720758354293053111T_VEBT: ( fun_VE6757229793580780342T_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE7029150624388687525_VEBTi: ( fun_VE712148701811721852_VEBTi * list_VEBT_VEBT ) > list_VEBT_VEBTi ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( fun_VE3165209603783583043T_VEBT * list_VEBT_VEBT ) > list_VEBT_VEBT ).

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,
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tff(sy_c_List_Olist_Oset_001t__HOL__Obool,type,
    set_bool2: fun_li6651299936896521309t_bool ).

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_It__HOL__Obool_J,type,
    set_list_bool2: list_list_bool > set_list_bool ).

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_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    set_Pr920681315882439344nteger: list_P5578671422887162913nteger > set_Pr4811707699266497531nteger ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__HOL__Obool_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__HOL__Obool_Mt__VEBT____Definitions__OVEBT_J,type,
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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__Nat__Onat_Mt__HOL__Obool_J,type,
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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__Nat__Onat_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr5984661752051438084T_VEBT: list_P5647936690300460905T_VEBT > set_Pr6167073792073659919T_VEBT ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__HOL__Obool_J,type,
    set_Pr3131006541957657995l_bool: list_P3020538015149218416l_bool > set_Pr2904644018331178518l_bool ).

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__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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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__Rat__Orat,type,
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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__HOL__Obool_J,type,
    set_set_bool2: list_set_bool > set_set_bool ).

tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Int__Oint_J,type,
    set_set_int2: list_set_int > set_set_int ).

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__Set__Oset_It__Real__Oreal_J,type,
    set_set_real2: list_set_real > set_set_real ).

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_Osize__list_001t__HOL__Obool,type,
    size_list_bool: ( fun_bool_nat * list_bool ) > nat ).

tff(sy_c_List_Olist_Osize__list_001t__Int__Oint,type,
    size_list_int: ( fun_int_nat * list_int ) > nat ).

tff(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
    size_list_nat: fun_nat_nat > fun_list_nat_nat ).

tff(sy_c_List_Olist_Osize__list_001t__Real__Oreal,type,
    size_list_real: ( fun_real_nat * list_real ) > nat ).

tff(sy_c_List_Olist_Osize__list_001t__Set__Oset_It__Nat__Onat_J,type,
    size_list_set_nat: ( fun_set_nat_nat * list_set_nat ) > nat ).

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

tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

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

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

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

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J,type,
    list_u9160741145793889621l_bool: ( list_P2722043638750504429l_bool * nat * produc2804947525362035943l_bool ) > list_P2722043638750504429l_bool ).

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

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

tff(sy_c_List_Olist__update_001t__VEBT____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 ) > fun_VE3114949619472531401T_VEBT ).

tff(sy_c_List_On__lists_001t__HOL__Obool,type,
    n_lists_bool: ( nat * list_bool ) > list_list_bool ).

tff(sy_c_List_On__lists_001t__Int__Oint,type,
    n_lists_int: ( nat * list_int ) > list_list_int ).

tff(sy_c_List_On__lists_001t__Nat__Onat,type,
    n_lists_nat: ( nat * list_nat ) > list_list_nat ).

tff(sy_c_List_On__lists_001t__Real__Oreal,type,
    n_lists_real: ( nat * list_real ) > list_list_real ).

tff(sy_c_List_On__lists_001t__VEBT____Definitions__OVEBT,type,
    n_lists_VEBT_VEBT: ( nat * list_VEBT_VEBT ) > list_list_VEBT_VEBT ).

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

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

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

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

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

tff(sy_c_List_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    nth_option_nat: list_option_nat > fun_nat_option_nat ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    nth_Pr2304437835452373666nteger: ( list_P5578671422887162913nteger * nat ) > produc8923325533196201883nteger ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J,type,
    nth_Pr2856333732135533678l_bool: ( list_P2722043638750504429l_bool * nat ) > produc2804947525362035943l_bool ).

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

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

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    nth_Pr8326237132889035090at_num: ( list_P1726324292696863441at_num * nat ) > product_prod_nat_num ).

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__VEBT____BuildupMemImp__OVEBTi_Mt__HOL__Obool_J,type,
    nth_Pr1878200693239305296i_bool: ( list_P4809559687441107495i_bool * nat ) > produc4444102050130342551i_bool ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    nth_Pr6911489093701683181Ti_nat: ( list_P659468882601404396Ti_nat * nat ) > produc3881548065746020326Ti_nat ).

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__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    nth_Pr6842391030413306568T_real: ( list_P2623026923184700063T_real * nat ) > produc5170161368751668367T_real ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr316670251186196177_VEBTi: ( list_P735349106241217576_VEBTi * nat ) > produc3625547720036274456_VEBTi ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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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__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_Oproduct_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc8792966785426426881nteger: ( list_Code_integer * list_Code_integer ) > list_P5578671422887162913nteger ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__HOL__Obool,type,
    product_bool_bool: ( list_bool * list_bool ) > list_P2722043638750504429l_bool ).

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

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Nat__Onat,type,
    product_bool_nat: ( list_bool * list_nat ) > list_P18149847613930150ol_nat ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Real__Oreal,type,
    product_bool_real: ( list_bool * list_real ) > list_P8975979458798387586l_real ).

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

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__HOL__Obool,type,
    product_nat_bool: ( list_nat * list_bool ) > list_P6781292141273955916t_bool ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: ( list_nat * list_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
    product_nat_num: ( list_nat * list_num ) > list_P1726324292696863441at_num ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Real__Oreal,type,
    product_nat_real: ( list_nat * list_real ) > list_P3644420460460130531t_real ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__HOL__Obool,type,
    product_real_bool: ( list_real * list_bool ) > list_P3020538015149218416l_bool ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Int__Oint,type,
    product_real_int: ( list_real * list_int ) > list_P4344331454722006975al_int ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    product_real_nat: ( list_real * list_nat ) > list_P6834414599653733731al_nat ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
    product_real_real: ( list_real * list_real ) > list_P8689742595348180415l_real ).

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

tff(sy_c_List_Oproduct__lists_001t__HOL__Obool,type,
    product_lists_bool: list_list_bool > list_list_bool ).

tff(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
    product_lists_int: list_list_int > list_list_int ).

tff(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
    product_lists_nat: list_list_nat > list_list_nat ).

tff(sy_c_List_Oproduct__lists_001t__Real__Oreal,type,
    product_lists_real: list_list_real > list_list_real ).

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__HOL__Obool,type,
    remove1_bool: ( bool * list_bool ) > list_bool ).

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__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J,type,
    remove4234689833455435907l_bool: ( produc2804947525362035943l_bool * list_P2722043638750504429l_bool ) > list_P2722043638750504429l_bool ).

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__Set__Oset_It__Nat__Onat_J,type,
    remove1_set_nat: ( set_nat * list_set_nat ) > list_set_nat ).

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_001t__HOL__Obool,type,
    removeAll_bool: ( bool * list_bool ) > list_bool ).

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

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

tff(sy_c_List_OremoveAll_001t__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__Code____Numeral__Ointeger,type,
    replic7707675349574490269nteger: ( nat * code_integer ) > list_Code_integer ).

tff(sy_c_List_Oreplicate_001t__HOL__Obool,type,
    replicate_bool: ( nat * bool ) > list_bool ).

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

tff(sy_c_List_Oreplicate_001t__List__Olist_It__HOL__Obool_J,type,
    replicate_list_bool: ( nat * list_bool ) > list_list_bool ).

tff(sy_c_List_Oreplicate_001t__List__Olist_It__Int__Oint_J,type,
    replicate_list_int: ( nat * list_int ) > list_list_int ).

tff(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
    replicate_list_nat: ( nat * list_nat ) > list_list_nat ).

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

tff(sy_c_List_Oreplicate_001t__Num__Onum,type,
    replicate_num: ( nat * num ) > list_num ).

tff(sy_c_List_Oreplicate_001t__Rat__Orat,type,
    replicate_rat: ( nat * rat ) > list_rat ).

tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
    replicate_real: nat > fun_real_list_real ).

tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: ( nat * set_nat ) > list_set_nat ).

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

tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
    replicate_VEBT_VEBT: nat > fun_VE3114949619472531401T_VEBT ).

tff(sy_c_List_Osorted__wrt_001t__Code____Numeral__Ointeger,type,
    sorted710888440204495920nteger: ( fun_Co8310303586393106575r_bool * list_Code_integer ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__HOL__Obool,type,
    sorted_wrt_bool: fun_bo1841576699660343469l_bool > fun_list_bool_bool ).

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_bool > fun_list_int_bool ).

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

tff(sy_c_List_Osorted__wrt_001t__Num__Onum,type,
    sorted_wrt_num: ( fun_num_fun_num_bool * list_num ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J,type,
    sorted378674801997687508l_bool: ( fun_Pr2956839735870039981l_bool * list_P2722043638750504429l_bool ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__Rat__Orat,type,
    sorted_wrt_rat: ( fun_rat_fun_rat_bool * list_rat ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__Real__Oreal,type,
    sorted_wrt_real: fun_re1255401033344146673l_bool > fun_list_real_bool ).

tff(sy_c_List_Osorted__wrt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    sorted9206477368072086664_VEBTi: ( fun_VE6376819126223186655i_bool * list_VEBT_VEBTi ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__VEBT____Definitions__OVEBT,type,
    sorted_wrt_VEBT_VEBT: ( fun_VE5877677516038742509T_bool * list_VEBT_VEBT ) > $o ).

tff(sy_c_List_Osubseqs_001t__HOL__Obool,type,
    subseqs_bool: list_bool > list_list_bool ).

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_001t__Code____Numeral__Ointeger,type,
    take_Code_integer: ( nat * list_Code_integer ) > list_Code_integer ).

tff(sy_c_List_Otake_001t__HOL__Obool,type,
    take_bool: ( nat * list_bool ) > list_bool ).

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__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__rel,type,
    upto_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_List_Ozip_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    zip_Co3543743374963494515nteger: ( list_Code_integer * list_Code_integer ) > list_P5578671422887162913nteger ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__HOL__Obool,type,
    zip_bool_bool: ( list_bool * list_bool ) > list_P2722043638750504429l_bool ).

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

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__Nat__Onat,type,
    zip_bool_nat: ( list_bool * list_nat ) > list_P18149847613930150ol_nat ).

tff(sy_c_List_Ozip_001t__HOL__Obool_001t__Real__Oreal,type,
    zip_bool_real: ( list_bool * list_real ) > list_P8975979458798387586l_real ).

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

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

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

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

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__HOL__Obool,type,
    zip_nat_bool: ( list_nat * list_bool ) > list_P6781292141273955916t_bool ).

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__Nat__Onat_001t__Num__Onum,type,
    zip_nat_num: ( list_nat * list_num ) > list_P1726324292696863441at_num ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__Real__Oreal,type,
    zip_nat_real: ( list_nat * list_real ) > list_P3644420460460130531t_real ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    zip_nat_VEBT_VEBT: ( list_nat * list_VEBT_VEBT ) > list_P5647936690300460905T_VEBT ).

tff(sy_c_List_Ozip_001t__Real__Oreal_001t__HOL__Obool,type,
    zip_real_bool: ( list_real * list_bool ) > list_P3020538015149218416l_bool ).

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__VEBT____BuildupMemImp__OVEBTi_001t__HOL__Obool,type,
    zip_VEBT_VEBTi_bool: ( list_VEBT_VEBTi * list_bool ) > list_P4809559687441107495i_bool ).

tff(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    zip_VEBT_VEBTi_nat: ( list_VEBT_VEBTi * list_nat ) > list_P659468882601404396Ti_nat ).

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_001t__HOL__Obool,type,
    zip_VEBT_VEBT_bool: ( list_VEBT_VEBT * list_bool ) > list_P44034876932920720T_bool ).

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

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

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__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_Map_Ograph_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    graph_5282091004195177018nteger: fun_Co8029881127924365703nteger > set_Pr4811707699266497531nteger ).

tff(sy_c_Map_Ograph_001t__Int__Oint_001t__Int__Oint,type,
    graph_int_int: fun_int_option_int > set_Pr958786334691620121nt_int ).

tff(sy_c_Map_Ograph_001t__Nat__Onat_001t__Nat__Onat,type,
    graph_nat_nat: fun_nat_option_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_Map_Ograph_001t__Nat__Onat_001t__Num__Onum,type,
    graph_nat_num: fun_nat_option_num > set_Pr6200539531224447659at_num ).

tff(sy_c_Map_Ograph_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    graph_VEBT_VEBT_nat: fun_VE8174548813682830439on_nat > set_Pr7556676689462069481BT_nat ).

tff(sy_c_Map_Omap__upds_001t__HOL__Obool_001t__HOL__Obool,type,
    map_upds_bool_bool: ( fun_bool_option_bool * list_bool * list_bool ) > fun_bool_option_bool ).

tff(sy_c_Map_Omap__upds_001t__HOL__Obool_001t__Int__Oint,type,
    map_upds_bool_int: ( fun_bool_option_int * list_bool * list_int ) > fun_bool_option_int ).

tff(sy_c_Map_Omap__upds_001t__HOL__Obool_001t__Nat__Onat,type,
    map_upds_bool_nat: ( fun_bool_option_nat * list_bool * list_nat ) > fun_bool_option_nat ).

tff(sy_c_Map_Omap__upds_001t__HOL__Obool_001t__Num__Onum,type,
    map_upds_bool_num: ( fun_bool_option_num * list_bool * list_num ) > fun_bool_option_num ).

tff(sy_c_Map_Omap__upds_001t__HOL__Obool_001t__Real__Oreal,type,
    map_upds_bool_real: ( fun_bool_option_real * list_bool * list_real ) > fun_bool_option_real ).

tff(sy_c_Map_Omap__upds_001t__HOL__Obool_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_up3574140571279353302_VEBTi: ( fun_bo5052482999933817007_VEBTi * list_bool * list_VEBT_VEBTi ) > fun_bo5052482999933817007_VEBTi ).

tff(sy_c_Map_Omap__upds_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    map_up5237471190051075923T_VEBT: ( fun_bo8890943138197189862T_VEBT * list_bool * list_VEBT_VEBT ) > fun_bo8890943138197189862T_VEBT ).

tff(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Int__Oint,type,
    map_upds_int_int: ( fun_int_option_int * list_int * list_int ) > fun_int_option_int ).

tff(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Nat__Onat,type,
    map_upds_int_nat: ( fun_int_option_nat * list_int * list_nat ) > fun_int_option_nat ).

tff(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Num__Onum,type,
    map_upds_int_num: ( fun_int_option_num * list_int * list_num ) > fun_int_option_num ).

tff(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    map_up4628693354496902621at_nat: ( fun_in7291707798943782516at_nat * list_int * list_P6011104703257516679at_nat ) > fun_in7291707798943782516at_nat ).

tff(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Real__Oreal,type,
    map_upds_int_real: ( fun_int_option_real * list_int * list_real ) > fun_int_option_real ).

tff(sy_c_Map_Omap__upds_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_up2395700066158073305_VEBTi: ( fun_in2850175258336333102_VEBTi * list_int * list_VEBT_VEBTi ) > fun_in2850175258336333102_VEBTi ).

tff(sy_c_Map_Omap__upds_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    map_up3480207007320876304T_VEBT: ( fun_in6246267328955250983T_VEBT * list_int * list_VEBT_VEBT ) > fun_in6246267328955250983T_VEBT ).

tff(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__HOL__Obool,type,
    map_upds_nat_bool: ( fun_nat_option_bool * list_nat * list_bool ) > fun_nat_option_bool ).

tff(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Int__Oint,type,
    map_upds_nat_int: ( fun_nat_option_int * list_nat * list_int ) > fun_nat_option_int ).

tff(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__HOL__Obool,type,
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tff(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__Int__Oint,type,
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tff(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__Nat__Onat,type,
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tff(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__Num__Onum,type,
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tff(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Map_Omap__upds_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
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tff(sy_c_Map_Omap__upds_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_Map_Omap__upds_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_Map_Oran_001t__Nat__Onat_001t__HOL__Obool,type,
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tff(sy_c_Map_Oran_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Map_Oran_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Map_Oran_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_Map_Oran_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Map_Oran_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Map_Oran_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Map_Orestrict__map_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_Map_Orestrict__map_001t__HOL__Obool_001t__Nat__Onat,type,
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tff(sy_c_Map_Orestrict__map_001t__HOL__Obool_001t__Num__Onum,type,
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tff(sy_c_Map_Orestrict__map_001t__HOL__Obool_001t__Real__Oreal,type,
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tff(sy_c_Map_Orestrict__map_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Map_Orestrict__map_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_Map_Orestrict__map_001t__Int__Oint_001t__Num__Onum,type,
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tff(sy_c_Map_Orestrict__map_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Map_Orestrict__map_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__HOL__Obool,type,
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tff(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__Int__Oint,type,
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tff(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__Nat__Onat,type,
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tff(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__Num__Onum,type,
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tff(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Map_Orestrict__map_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
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tff(sy_c_Map_Orestrict__map_001t__Set__Oset_It__Nat__Onat_J_001t__Num__Onum,type,
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tff(sy_c_Map_Orestrict__map_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_Map_Orestrict__map_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_Map_Orestrict__map_001t__VEBT____Definitions__OVEBT_001t__Num__Onum,type,
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tff(sy_c_Misc_Omerge_001t__HOL__Obool,type,
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tff(sy_c_Misc_Omerge_001t__Int__Oint,type,
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tff(sy_c_Misc_Omerge_001t__Nat__Onat,type,
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tff(sy_c_Misc_Omerge_001t__Num__Onum,type,
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tff(sy_c_Misc_Omerge_001t__Rat__Orat,type,
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tff(sy_c_Misc_Omerge_001t__Real__Oreal,type,
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tff(sy_c_Misc_Omergesort__remdups_001t__HOL__Obool,type,
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tff(sy_c_Misc_Omergesort__remdups_001t__Int__Oint,type,
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tff(sy_c_Misc_Omergesort__remdups_001t__Nat__Onat,type,
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tff(sy_c_Misc_Omergesort__remdups_001t__Num__Onum,type,
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tff(sy_c_Misc_Omergesort__remdups_001t__Rat__Orat,type,
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tff(sy_c_Misc_Omergesort__remdups_001t__Real__Oreal,type,
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tff(sy_c_Misc_Oslice_001t__HOL__Obool,type,
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tff(sy_c_Misc_Oslice_001t__Int__Oint,type,
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tff(sy_c_Misc_Oslice_001t__Nat__Onat,type,
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tff(sy_c_Misc_Oslice_001t__Real__Oreal,type,
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tff(sy_c_Misc_Oslice_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Misc_Oslice_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__HOL__Obool,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,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__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__Option__Ooption_It__Int__Oint_J,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_Omax_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HOL__Obool_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Assertions__Oassn,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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    produc101793102246108661omplex: ( complex * complex ) > produc4411394909380815293omplex ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__HOL__Obool,type,
    produc8297744024737662807l_bool: bool > fun_bo8995055541810095699l_bool ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Nat__Onat,type,
    produc8874003862241262736ol_nat: bool > fun_na6010711327783111523ol_nat ).

tff(sy_c_Product__Type_OPair_001t__HOL__Obool_001t__Real__Oreal,type,
    produc7587643690356850540l_real: bool > fun_re4819860838828136987l_real ).

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

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > fun_in4211310926729683942nt_int ).

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__Nat__Onat_001t__HOL__Obool,type,
    produc3913204875133826030t_bool: nat > fun_bo5028639161814991974t_bool ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
    product_Pair_nat_int: ( nat * int ) > product_prod_nat_int ).

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

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
    product_Pair_nat_num: nat > fun_nu3130108178629567498at_num ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: ( nat * product_prod_nat_nat ) > produc7248412053542808358at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    produc1195630363706982562at_num: ( nat * product_prod_nat_num ) > produc2963631642982155120at_num ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
    produc7837566107596912789t_real: nat > fun_re5596339207225094666t_real ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc2649746096677893406_VEBTi: ( nat * vEBT_VEBTi ) > produc214224863196444774_VEBTi ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc599794634098209291T_VEBT: nat > fun_VE1635941959115973792T_VEBT ).

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

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

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

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__HOL__Obool,type,
    produc3350604258665657362l_bool: real > fun_bo5584942545454265226l_bool ).

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__Nat__Onat,type,
    produc3181502643871035669al_nat: real > fun_na3535369430382908390al_nat ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: real > fun_re5118428557197245286l_real ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc6931449550656315951T_VEBT: real > fun_VE3519909879894771012T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__HOL__Obool,type,
    produc7282046070707253577i_bool: ( vEBT_VEBTi * bool ) > produc4444102050130342551i_bool ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    produc7192665754729510430Ti_nat: ( vEBT_VEBTi * nat ) > produc3881548065746020326Ti_nat ).

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_001t__HOL__Obool,type,
    produc3035007008238225530T_bool: vEBT_VEBT > fun_bo4254975943616364022T_bool ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: vEBT_VEBT > fun_in9220260400085560248BT_int ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: vEBT_VEBT > fun_na4013023315904204544BT_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8117437818029410057T_real: vEBT_VEBT > fun_re4081685989952073528T_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 > fun_VE6757229793580780342T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc2004789812759622991on_nat: ( fun_nat_fun_nat_bool * produc4953844613479565601on_nat ) > produc3700678068921409119on_nat ).

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

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    produc460779904311030791at_num: ( fun_nat_fun_num_num * produc2963631642982155120at_num ) > produc8114805097830739791at_num ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc8433281988922931063on_num: ( fun_num_fun_num_bool * produc3447558737645232053on_num ) > produc6167219322466932359on_num ).

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc5961576194287291153at_nat: ( fun_Pr248683899324676269t_bool * produc6121120109295599847at_nat ) > produc533602882674032673at_nat ).

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

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__HOL__Obool_001t__HOL__Obool_001t__HOL__Obool,type,
    produc8645476572806212523l_bool: fun_bo1841576699660343469l_bool > fun_Pr4298194452408323251l_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool,type,
    produc8560705615207772305t_bool: fun_int_fun_int_bool > fun_Pr803113274115125473t_bool ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: ( fun_int_fun_int_int * product_prod_int_int ) > int ).

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc6842872674320459806at_nat: fun_nat_fun_nat_nat > fun_Pr2602117396471556196at_nat ).

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    produc478579273971653890on_num: ( fun_na7595719129879163854on_num * product_prod_nat_num ) > option_num ).

tff(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001t__HOL__Obool,type,
    produc3346841887225286771t_bool: ( set_nat * set_bool ) > set_Pr4681767519595310322t_bool ).

tff(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001t__Real__Oreal,type,
    produc6734486746367016272t_real: ( set_nat * set_real ) > set_Pr320017278500174781t_real ).

tff(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc1441754205559175824T_VEBT: ( set_nat * set_VEBT_VEBT ) > set_Pr6167073792073659919T_VEBT ).

tff(sy_c_Product__Type_Oproduct_001t__Real__Oreal_001t__HOL__Obool,type,
    produc9024588101255028247l_bool: ( set_real * set_bool ) > set_Pr2904644018331178518l_bool ).

tff(sy_c_Product__Type_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    produc2078423282641139152al_nat: ( set_real * set_nat ) > set_Pr3510011417693777981al_nat ).

tff(sy_c_Product__Type_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
    produc2998788613821421740l_real: ( set_real * set_real ) > set_Pr6218003697084177305l_real ).

tff(sy_c_Product__Type_Oproduct_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc7150050738623674420T_VEBT: ( set_real * set_VEBT_VEBT ) > set_Pr6019664923565264691T_VEBT ).

tff(sy_c_Product__Type_Oproduct_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    produc5189178589784285493T_bool: ( set_VEBT_VEBT * set_bool ) > set_Pr2229214029182756970T_bool ).

tff(sy_c_Product__Type_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8336039005996768526T_real: ( set_VEBT_VEBT * set_real ) > set_Pr7765410600122031685T_real ).

tff(sy_c_Product__Type_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc3886929683002245970T_VEBT: ( set_VEBT_VEBT * set_VEBT_VEBT ) > set_Pr6192946355708809607T_VEBT ).

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

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

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_complex_real ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__HOL__Obool,type,
    refine4616567020171210338s_bool: ( heap_Time_Heap_bool * heap_Time_Heap_bool ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__HOL__Obool_J,type,
    refine3902108050975373032t_bool: ( heap_T5178755288483056679t_bool * heap_T5178755288483056679t_bool ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    refine3700189196150522554_VEBTi: ( heap_T4980287057938770641_VEBTi * heap_T4980287057938770641_VEBTi ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__Nat__Onat,type,
    refine1365783493865988805es_nat: ( heap_Time_Heap_nat * heap_Time_Heap_nat ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__Option__Ooption_It__Nat__Onat_J,type,
    refine7594492741263601813on_nat: ( heap_T2636463487746394924on_nat * heap_T2636463487746394924on_nat ) > $o ).

tff(sy_c_Refine__Imp__Hol_Orefines_001t__VEBT____BuildupMemImp__OVEBTi,type,
    refine5565527176597971370_VEBTi: ( heap_T8145700208782473153_VEBTi * heap_T8145700208782473153_VEBTi ) > $o ).

tff(sy_c_Relation_OId__on_001t__Code____Numeral__Ointeger,type,
    id_on_Code_integer: set_Code_integer > set_Pr4811707699266497531nteger ).

tff(sy_c_Relation_OId__on_001t__Complex__Ocomplex,type,
    id_on_complex: set_complex > set_Pr5085853215250843933omplex ).

tff(sy_c_Relation_OId__on_001t__HOL__Obool,type,
    id_on_bool: set_bool > set_Pr8621012493645324487l_bool ).

tff(sy_c_Relation_OId__on_001t__Int__Oint,type,
    id_on_int: set_int > set_Pr958786334691620121nt_int ).

tff(sy_c_Relation_OId__on_001t__Nat__Onat,type,
    id_on_nat: set_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_Relation_OId__on_001t__Real__Oreal,type,
    id_on_real: set_real > set_Pr6218003697084177305l_real ).

tff(sy_c_Relation_OId__on_001t__VEBT____Definitions__OVEBT,type,
    id_on_VEBT_VEBT: set_VEBT_VEBT > set_Pr6192946355708809607T_VEBT ).

tff(sy_c_Relation_Ototal__on_001t__Code____Numeral__Ointeger,type,
    total_8516208958838685657nteger: ( set_Code_integer * set_Pr4811707699266497531nteger ) > $o ).

tff(sy_c_Relation_Ototal__on_001t__HOL__Obool,type,
    total_on_bool: ( set_bool * set_Pr8621012493645324487l_bool ) > $o ).

tff(sy_c_Relation_Ototal__on_001t__Int__Oint,type,
    total_on_int: ( set_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_Relation_Ototal__on_001t__Nat__Onat,type,
    total_on_nat: ( set_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Relation_Ototal__on_001t__Real__Oreal,type,
    total_on_real: ( set_real * set_Pr6218003697084177305l_real ) > $o ).

tff(sy_c_Relation_Ototal__on_001t__VEBT____Definitions__OVEBT,type,
    total_on_VEBT_VEBT: ( set_VEBT_VEBT * set_Pr6192946355708809607T_VEBT ) > $o ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > fun_real_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_bool_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: fun_bool_nat ).

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

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

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

tff(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: fun_Co8822979462182099027r_bool > set_Code_integer ).

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

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    is_singleton_nat: fun_set_nat_bool ).

tff(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
    is_singleton_real: fun_set_real_bool ).

tff(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
    is_singleton_set_nat: fun_set_set_nat_bool ).

tff(sy_c_Set_Ois__singleton_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Set_Othe__elem_001t__HOL__Obool,type,
    the_elem_bool: fun_set_bool_bool ).

tff(sy_c_Set_Othe__elem_001t__Int__Oint,type,
    the_elem_int: fun_set_int_int ).

tff(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: fun_set_nat_nat ).

tff(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
    the_elem_real: set_real > real ).

tff(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
    the_elem_set_nat: set_set_nat > set_nat ).

tff(sy_c_Set_Othe__elem_001t__VEBT____Definitions__OVEBT,type,
    the_elem_VEBT_VEBT: set_VEBT_VEBT > vEBT_VEBT ).

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

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

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Num__Onum,type,
    set_fo8365102181078989356at_num: ( fun_nat_fun_num_num * nat ) > fun_nat_fun_num_num ).

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

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
    set_fo3699595496184130361el_nat: fun_Pr4617804334418790189t_bool ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Num__Onum,type,
    set_fo256927282339908995el_num: fun_Pr7353992663003996845m_bool ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: int > fun_int_set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__HOL__Obool,type,
    set_or8576162586734908840n_bool: bool > fun_bool_set_bool ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: int > fun_int_set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: fun_na6128946301132549280et_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
    set_or1222409239386451017an_num: ( num * num ) > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Rat__Orat,type,
    set_or4029947393144176647an_rat: ( rat * rat ) > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
    set_or66887138388493659n_real: real > fun_real_set_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Int__Oint_J,type,
    set_or8585797421378605585et_int: ( set_int * set_int ) > set_set_int ).

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

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
    set_ord_atMost_int: fun_int_set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: fun_int_set_int ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__divide_001t__Int__Oint,type,
    signed6714573509424544716de_int: fun_int_fun_int_int ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
    signed6292675348222524329lo_int: fun_int_fun_int_int ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__HOL__Obool,type,
    time_TBOUND_bool: heap_Time_Heap_bool > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__HOL__Obool_J,type,
    time_T4182777596009358560y_bool: heap_T6374562886178639292y_bool > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__Int__Oint_J,type,
    time_T8667231999468956601ay_int: heap_T1346037964561226099ay_int > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__Nat__Onat_J,type,
    time_T3621710982123377501ay_nat: heap_T3836121109492952855ay_nat > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__HOL__Obool_J,type,
    time_T5840735555483777133t_bool: heap_T5178755288483056679t_bool > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Nat__Onat_J,type,
    time_TBOUND_list_nat: heap_T290393402774840812st_nat > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    time_T3808005469503390304on_nat: heap_T5317711798761887292on_nat > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T8149879359713347829_VEBTi: heap_T4980287057938770641_VEBTi > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Nat__Onat,type,
    time_TBOUND_nat: heap_Time_Heap_nat > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_T8353473612707095248on_nat: heap_T2636463487746394924on_nat > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_T5737551269749752165_VEBTi: heap_T8145700208782473153_VEBTi > fun_nat_bool ).

tff(sy_c_Time__Reasoning_Ofails_001t__HOL__Obool,type,
    time_fails_bool: ( heap_Time_Heap_bool * heap_e7401611519738050253t_unit ) > $o ).

tff(sy_c_Time__Reasoning_Ofails_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_f8309030937046825446_VEBTi: ( heap_T8145700208782473153_VEBTi * heap_e7401611519738050253t_unit ) > $o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__HOL__Obool,type,
    time_htt_bool: ( assn * heap_Time_Heap_bool * fun_bool_assn ) > fun_nat_bool ).

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

tff(sy_c_Time__Reasoning_Ohtt_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_htt_option_nat: ( assn * heap_T2636463487746394924on_nat * fun_option_nat_assn ) > fun_nat_bool ).

tff(sy_c_Time__Reasoning_Ohtt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_htt_VEBT_VEBTi: ( assn * heap_T8145700208782473153_VEBTi * fun_VEBT_VEBTi_assn ) > fun_nat_bool ).

tff(sy_c_Time__Reasoning_Otime_001t__HOL__Obool,type,
    time_time_bool: ( heap_Time_Heap_bool * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__HOL__Obool_J,type,
    time_time_array_bool: ( heap_T6374562886178639292y_bool * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__Int__Oint_J,type,
    time_time_array_int: ( heap_T1346037964561226099ay_int * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__Nat__Onat_J,type,
    time_time_array_nat: ( heap_T3836121109492952855ay_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t3534373299052942712_VEBTi: ( heap_T4980287057938770641_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Nat__Onat,type,
    time_time_nat: ( heap_Time_Heap_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_time_option_nat: ( heap_T2636463487746394924on_nat * heap_e7401611519738050253t_unit ) > nat ).

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

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

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

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > fun_real_real ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Transitive__Closure_Otrancl_001t__Code____Numeral__Ointeger,type,
    transi6870300401645067644nteger: set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger ).

tff(sy_c_Transitive__Closure_Otrancl_001t__HOL__Obool,type,
    transi8112060559632522936l_bool: set_Pr8621012493645324487l_bool > set_Pr8621012493645324487l_bool ).

tff(sy_c_Transitive__Closure_Otrancl_001t__Int__Oint,type,
    transi6261509568448316235cl_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

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

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__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_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_Pr817965127378462947t_bool ).

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_Pr817965127378462947t_bool ).

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_VE5877677516038742509T_bool ).

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_Pr817965127378462947t_bool ).

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_Pr817965127378462947t_bool ).

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_VE5877677516038742509T_bool ).

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_VE5877677516038742509T_bool ).

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_Pr817965127378462947t_bool ).

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_Pr817965127378462947t_bool ).

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_Pr817965127378462947t_bool ).

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_Pr817965127378462947t_bool ).

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_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_bool ).

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_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun_nat_fun_nat_bool ).

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_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__HOL__Obool,type,
    vEBT_V6044891547498605301i_bool: ( nat * heap_Time_Heap_bool ) > heap_T5178755288483056679t_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: ( nat * heap_Time_Heap_nat ) > heap_T290393402774840812st_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_V792416675989592002on_nat: ( nat * heap_T2636463487746394924on_nat ) > heap_T5317711798761887292on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: ( nat * heap_T8145700208782473153_VEBTi ) > heap_T4980287057938770641_VEBTi ).

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_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__HOL__Obool_J,type,
    vEBT_c4579152981172362421p_bool: ( fun_op3392518339485081117p_bool * fun_bo7602104967519523767p_bool * vEBT_VEBTi ) > heap_Time_Heap_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( fun_op5127434927081926045_VEBTi * fun_bo851793448214054487_VEBTi * vEBT_VEBTi ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun_VE905575822219544865i_assn ).

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__memberi,type,
    vEBT_vebt_memberi: ( vEBT_VEBTi * nat ) > heap_Time_Heap_bool ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: bool > fun_bool_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_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: fun_VE9028255658833972685et_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_bool ).

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_Pr817965127378462947t_bool ).

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_Pr817965127378462947t_bool ).

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_Pr817965127378462947t_bool ).

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_VE5877677516038742509T_bool ).

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_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__HOL__Obool_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6372884366951783724_VEBTi: ( set_nat * fun_bo7535889243153246212i_assn * list_bool * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2992182275864018813T_VEBT: ( set_nat * fun_bo5394993412183903963T_assn * list_bool * list_VEBT_VEBT ) > 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__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__Nat__Onat,type,
    vEBT_L234762979517870878al_nat: ( set_nat * fun_re574865750389657758t_assn * list_real * list_nat ) > 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_001t__HOL__Obool,type,
    vEBT_L2302322349616425472i_bool: ( set_nat * fun_VE3808733824349961398l_assn * list_VEBT_VEBTi * list_bool ) > 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__HOL__Obool,type,
    vEBT_L6176004506706552451T_bool: ( set_nat * fun_VE1323945509527881313l_assn * list_VEBT_VEBT * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8648204552663881920BT_int: ( set_nat * fun_VE123837653852366616t_assn * list_VEBT_VEBT * list_int ) > 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__Real__Oreal,type,
    vEBT_L4281036506115550016T_real: ( set_nat * fun_VE6910866152645720856l_assn * list_VEBT_VEBT * list_real ) > 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_001t__HOL__Obool_001t__HOL__Obool,type,
    vEBT_L6562251693209944067l_bool: ( fun_bo8172996160782635326l_assn * list_bool ) > fun_list_bool_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__Int__Oint,type,
    vEBT_L7518048937714687808ol_int: ( fun_bo3630985899495134459t_assn * list_bool * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__Nat__Onat,type,
    vEBT_L7520539408223738084ol_nat: ( fun_bo8438563643002758303t_assn * list_bool ) > fun_list_nat_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_L5912725412549479476on_nat: ( fun_bo1795969981296709615t_assn * list_bool ) > fun_li338670087962420949t_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__Real__Oreal,type,
    vEBT_L5107238702033023424l_real: ( fun_bo8885878543415060347l_assn * list_bool * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1689158095293668553_VEBTi: ( fun_bo7535889243153246212i_assn * list_bool ) > fun_li3352006551604695530i_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__Nat__Onat_001t__HOL__Obool,type,
    vEBT_L2559740421116301378t_bool: ( fun_na5136419689362302555l_assn * list_nat * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Int__Oint,type,
    vEBT_L8298612041380073281at_int: ( fun_nat_fun_int_assn * list_nat * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Nat__Onat,type,
    vEBT_L8301102511889123557at_nat: ( fun_nat_fun_nat_assn * list_nat * list_nat ) > assn ).

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

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__HOL__Obool,type,
    vEBT_L870199270341830246l_bool: ( fun_re2206215847901373695l_assn * list_real * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Int__Oint,type,
    vEBT_L1443519841834266653al_int: ( fun_re4990660043736809722t_assn * list_real * list_int ) > assn ).

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

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

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L9060850011106065574_VEBTi: ( fun_re560840401387337987i_assn * list_real * list_VEBT_VEBTi ) > 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_001t__HOL__Obool,type,
    vEBT_L5320321625355440422T_bool: ( fun_VE1323945509527881313l_assn * list_VEBT_VEBT ) > fun_list_bool_assn ).

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

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

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_L8010285020845282001on_nat: ( fun_VE6334927249922962060t_assn * list_VEBT_VEBT ) > fun_li338670087962420949t_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 ) > fun_li3352006551604695530i_assn ).

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

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: fun_VE9028255658833972685et_nat ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op7792301606653096457t_bool ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

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

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: fun_Pr4582292742318232305t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
    vEBT_V452583751252753300el_num: fun_Pr205781412008949125m_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V7235779383477046023at_nat: fun_Pr8292650585710461613t_bool ).

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: fun_VE8174548813682830439on_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: fun_VE8174548813682830439on_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE5877677516038742509T_bool ).

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

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: vEBT_VEBT > fun_nat_option_nat ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr817965127378462947t_bool ).

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_bool ).

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_bool ).

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_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun_VE5877677516038742509T_bool ).

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_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun_VE5877677516038742509T_bool ).

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__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: vEBT_VEBT > fun_nat_option_nat ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: fun_li8598321315282354825t_bool > fun_list_nat_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_bool > fun_nat_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: fun_Pr3099820731930151597t_bool > fun_Pr803113274115125473t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: fun_Pr248683899324676269t_bool > fun_Pr7757163693017731481t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: fun_Pr817965127378462947t_bool > fun_Pr2834072668759824765t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P4694664897847628410at_nat: ( fun_Pr4617804334418790189t_bool * produc9217583074051984497at_nat ) > $o ).

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    accp_P5526603486790338531on_nat: ( fun_Pr4582292742318232305t_bool * produc7004362771917097900on_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Wellfounded_Omeasure_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
    measure_int: fun_int_nat > set_Pr958786334691620121nt_int ).

tff(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: fun_nat_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    aa_assn_assn: ( fun_assn_assn * assn ) > assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__HOL__Obool,type,
    aa_assn_bool: ( fun_assn_bool * assn ) > bool ).

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__Assertions__Oassn_001t__fun_It__Assertions__Oassn_Mt__HOL__Obool_J,type,
    aa_ass5323576183532441186n_bool: ( fun_as7903617535776176121n_bool * assn ) > fun_assn_bool ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__HOL__Obool_Mt__Assertions__Oassn_J,type,
    aa_ass2376637134808068204l_assn: ( fun_as6968651184343528707l_assn * assn ) > fun_bool_assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
    aa_assn_fun_nat_assn: ( fun_as8240436605320707994t_assn * assn ) > fun_nat_assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Assertions__Oassn_J,type,
    aa_ass4112278883933122603t_assn: ( fun_as2043416024874063466t_assn * assn ) > fun_option_nat_assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J,type,
    aa_ass4389913640333755968i_assn: ( fun_as315122898794150911i_assn * assn ) > fun_VEBT_VEBTi_assn ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Option__Ooption_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Assertions__Oassn,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Num__Onum,type,
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tff(sy_c_aa_001t__Complex__Ocomplex_001t__Rat__Orat,type,
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tff(sy_c_aa_001t__List__Olist_It__Int__Oint_J_001t__fun_It__Nat__Onat_Mt__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Int__Oint_J_001t__fun_It__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_Mt__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__Assertions__Oassn,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__Heap__Oarray_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Assertions__Oassn_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__Heap__Oarray_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Heap__Oarray_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__Real__Oreal_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Nat__Onat_J_001t__fun_It__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_Mt__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J_001t__Assertions__Oassn,type,
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tff(sy_c_aa_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J_001t__fun_It__Heap__Oarray_It__Option__Ooption_It__Nat__Onat_J_J_Mt__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__HOL__Obool,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__Nat__Onat,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__HOL__Obool_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__HOL__Obool_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__HOL__Obool_J_Mt__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__HOL__Obool_J_Mt__fun_It__List__Olist_It__Real__Oreal_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__List__Olist_It__Real__Oreal_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Nat__Onat_J_Mt__fun_It__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_Mt__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Real__Oreal_J_Mt__HOL__Obool_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Real__Oreal_J_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Real__Oreal_J_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Real__Oreal_J_Mt__fun_It__List__Olist_It__Nat__Onat_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__Real__Oreal_J_Mt__fun_It__List__Olist_It__Real__Oreal_J_Mt__HOL__Obool_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__fun_It__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_Mt__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_Mt__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__fun_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__List__Olist_It__Real__Oreal_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__Real__Oreal_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Assertions__Oassn,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__fun_It__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__Assertions__Oassn_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__fun_It__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__fun_It__Int__Oint_Mt__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__Nat__Onat_Mt__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_J_J,type,
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tff(sy_c_aa_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__fun_It__List__Olist_It__Int__Oint_J_Mt__fun_It__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_Mt__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_J_J,type,
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tff(sy_c_fFalse,type,
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tff(sy_c_fNot,type,
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tff(sy_c_fTrue,type,
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tff(sy_c_fconj,type,
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tff(sy_c_fdisj,type,
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tff(sy_c_fequal_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_fequal_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
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tff(sy_c_member_001t__Nat__Onat,type,
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tff(sy_c_member_001t__Num__Onum,type,
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tff(sy_c_pp,type,
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tff(sy_v_tia____,type,
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% Relevant facts (11188)
tff(fact_0_valid__0__not,axiom,
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% valid_0_not
tff(fact_1_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_2_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_3_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2),S) ) ).

% deg_SUcn_Node
tff(fact_4__C5_C,axiom,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,v),aa_nat_nat(suc,zero_zero_nat),vg),vh)),na)) ).

% "5"
tff(fact_5_norm__pre__pure__iff,axiom,
    ! [P: assn,B: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q)
    <=> ( pp(B)
       => hoare_7629718768684598413on_nat(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_6_norm__pre__pure__iff,axiom,
    ! [P: assn,B: bool,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q)
    <=> ( pp(B)
       => hoare_2774773592644289642e_bool(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_7_norm__pre__pure__iff,axiom,
    ! [P: assn,B: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q)
    <=> ( pp(B)
       => hoare_1429296392585015714_VEBTi(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_8_norm__pre__pure__iff,axiom,
    ! [P: assn,B: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q)
    <=> ( pp(B)
       => hoare_3067605981109127869le_nat(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_9_merge__pure__star,axiom,
    ! [A: bool,B: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,A)),aa_bool_assn(pure_assn,B)) = aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,A),B)) ).

% merge_pure_star
tff(fact_10_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_11_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_12_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_13_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_14_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_15_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_16_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_17_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_18_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_19_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_20_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_21_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_22_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_23_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_24_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_25_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_26_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_27_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_28_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_29_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_30_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_31_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_32_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_33_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_34_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_35_norm__pre__pure__rule1,axiom,
    ! [B: bool,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ( pp(B)
       => hoare_7629718768684598413on_nat(P,F,Q) )
     => hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_36_norm__pre__pure__rule1,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( ( pp(B)
       => hoare_2774773592644289642e_bool(P,F,Q) )
     => hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_37_norm__pre__pure__rule1,axiom,
    ! [B: bool,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ( pp(B)
       => hoare_1429296392585015714_VEBTi(P,F,Q) )
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_38_norm__pre__pure__rule1,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ( pp(B)
       => hoare_3067605981109127869le_nat(P,F,Q) )
     => hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_39_pure__assn__eq__conv,axiom,
    ! [P: bool,Q: bool] :
      ( ( aa_bool_assn(pure_assn,P) = aa_bool_assn(pure_assn,Q) )
    <=> ( pp(P)
      <=> pp(Q) ) ) ).

% pure_assn_eq_conv
tff(fact_40_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_41_frame__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R: assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),C,aa_ass4112278883933122603t_assn(aTP_Lamm_aa(Q),R)) ) ).

% frame_rule
tff(fact_42_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,R: assn] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),C,aa_ass2376637134808068204l_assn(aTP_Lamm_ab(Q),R)) ) ).

% frame_rule
tff(fact_43_frame__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),C,aa_ass4389913640333755968i_assn(aTP_Lamm_ac(Q),R)) ) ).

% frame_rule
tff(fact_44_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,R: assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),C,aa_assn_fun_nat_assn(aTP_Lamm_ad(Q),R)) ) ).

% frame_rule
tff(fact_45_frame__rule__left,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R: assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),P),C,aa_ass4112278883933122603t_assn(aTP_Lamm_ae(Q),R)) ) ).

% frame_rule_left
tff(fact_46_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,R: assn] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),P),C,aa_ass2376637134808068204l_assn(aTP_Lamm_af(Q),R)) ) ).

% frame_rule_left
tff(fact_47_frame__rule__left,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),P),C,aa_ass4389913640333755968i_assn(aTP_Lamm_ag(Q),R)) ) ).

% frame_rule_left
tff(fact_48_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,R: assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),P),C,aa_assn_fun_nat_assn(aTP_Lamm_ah(Q),R)) ) ).

% frame_rule_left
tff(fact_49_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),R) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),R)) ).

% assn_times_assoc
tff(fact_50_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),P) ).

% assn_times_comm
tff(fact_51_star__assoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).

% star_assoc
tff(fact_52_star__aci_I2_J,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) ).

% star_aci(2)
tff(fact_53_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)) ).

% star_aci(3)
tff(fact_54_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)),B) ).

% assn_aci(10)
tff(fact_55_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => hoare_7629718768684598413on_nat(P,C,Q) ) ).

% is_hoare_triple
tff(fact_56_is__hoare__triple,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => hoare_2774773592644289642e_bool(P,C,Q) ) ).

% is_hoare_triple
tff(fact_57_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => hoare_1429296392585015714_VEBTi(P,C,Q) ) ).

% is_hoare_triple
tff(fact_58_is__hoare__triple,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => hoare_3067605981109127869le_nat(P,C,Q) ) ).

% is_hoare_triple
tff(fact_59_if__rule,axiom,
    ! [B: bool,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,G: heap_T2636463487746394924on_nat] :
      ( ( pp(B)
       => hoare_7629718768684598413on_nat(P,F,Q) )
     => ( ( ~ pp(B)
         => hoare_7629718768684598413on_nat(P,G,Q) )
       => hoare_7629718768684598413on_nat(P,if_Hea5867803462524415986on_nat(B,F,G),Q) ) ) ).

% if_rule
tff(fact_60_if__rule,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_bool,Q: fun_bool_assn,G: heap_Time_Heap_bool] :
      ( ( pp(B)
       => hoare_2774773592644289642e_bool(P,F,Q) )
     => ( ( ~ pp(B)
         => hoare_2774773592644289642e_bool(P,G,Q) )
       => hoare_2774773592644289642e_bool(P,if_Hea7696252282596813915p_bool(B,F,G),Q) ) ) ).

% if_rule
tff(fact_61_if__rule,axiom,
    ! [B: bool,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,G: heap_T8145700208782473153_VEBTi] :
      ( ( pp(B)
       => hoare_1429296392585015714_VEBTi(P,F,Q) )
     => ( ( ~ pp(B)
         => hoare_1429296392585015714_VEBTi(P,G,Q) )
       => hoare_1429296392585015714_VEBTi(P,if_Hea8453224502484754311_VEBTi(B,F,G),Q) ) ) ).

% if_rule
tff(fact_62_if__rule,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn,G: heap_Time_Heap_nat] :
      ( ( pp(B)
       => hoare_3067605981109127869le_nat(P,F,Q) )
     => ( ( ~ pp(B)
         => hoare_3067605981109127869le_nat(P,G,Q) )
       => hoare_3067605981109127869le_nat(P,if_Hea2662716070787841314ap_nat(B,F,G),Q) ) ) ).

% if_rule
tff(fact_63_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_64_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_65_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_66_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_67_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_68_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_69_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_70_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_71_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_72_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_73_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_74_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_75_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_76_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_77_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_78_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_79_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_80_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_81_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_82_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_83_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_84_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_85_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_86_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_87_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_88_lambda__zero,axiom,
    aTP_Lamm_ai = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_89_lambda__zero,axiom,
    aTP_Lamm_aj = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_90_lambda__zero,axiom,
    aTP_Lamm_ak = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_91_lambda__zero,axiom,
    aTP_Lamm_al = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_92_lambda__zero,axiom,
    aTP_Lamm_am = aa_com3342990543769683525omplex(times_times_complex,zero_zero_complex) ).

% lambda_zero
tff(fact_93_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_94_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_95_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X2) = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_96_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_option_nat(some_nat,X2) = aa_nat_option_nat(some_nat,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_97_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_option_num(some_num,X2) = aa_num_option_num(some_num,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_98_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_99_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_100_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_101_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_102_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_103_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_104_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D)) ) ).

% valid_eq2
tff(fact_105_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D)) ) ).

% valid_eq
tff(fact_106_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),collect_VEBT_VEBT(P)))
    <=> pp(aa_VEBT_VEBT_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_107_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(A),collect_real(P)))
    <=> pp(aa_real_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_108_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(member_complex(A),collect_complex(P)))
    <=> pp(aa_complex_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_109_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] :
      ( pp(aa_set_list_nat_bool(member_list_nat(A),collect_list_nat(P)))
    <=> pp(aa_list_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_110_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),collect_set_nat(P)))
    <=> pp(aa_set_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_111_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A),collect_nat(P)))
    <=> pp(aa_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_112_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A),collect_int(P)))
    <=> pp(aa_int_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_113_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] : collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_an,A2)) = A2 ).

% Collect_mem_eq
tff(fact_114_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_ao,A2)) = A2 ).

% Collect_mem_eq
tff(fact_115_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_ap(A2)) = A2 ).

% Collect_mem_eq
tff(fact_116_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_aq(A2)) = A2 ).

% Collect_mem_eq
tff(fact_117_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_ar,A2)) = A2 ).

% Collect_mem_eq
tff(fact_118_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_as,A2)) = A2 ).

% Collect_mem_eq
tff(fact_119_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_at,A2)) = A2 ).

% Collect_mem_eq
tff(fact_120_Collect__cong,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ! [X: complex] :
          ( pp(aa_complex_bool(P,X))
        <=> pp(aa_complex_bool(Q,X)) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_121_Collect__cong,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X: list_nat] :
          ( pp(aa_list_nat_bool(P,X))
        <=> pp(aa_list_nat_bool(Q,X)) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_122_Collect__cong,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X: set_nat] :
          ( pp(aa_set_nat_bool(P,X))
        <=> pp(aa_set_nat_bool(Q,X)) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_123_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X: nat] :
          ( pp(aa_nat_bool(P,X))
        <=> pp(aa_nat_bool(Q,X)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_124_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X: int] :
          ( pp(aa_int_bool(P,X))
        <=> pp(aa_int_bool(Q,X)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_125_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D)) ) ).

% valid_eq1
tff(fact_126_zero__reorient,axiom,
    ! [X3: complex] :
      ( ( zero_zero_complex = X3 )
    <=> ( X3 = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_127_zero__reorient,axiom,
    ! [X3: real] :
      ( ( zero_zero_real = X3 )
    <=> ( X3 = zero_zero_real ) ) ).

% zero_reorient
tff(fact_128_zero__reorient,axiom,
    ! [X3: rat] :
      ( ( zero_zero_rat = X3 )
    <=> ( X3 = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_129_zero__reorient,axiom,
    ! [X3: nat] :
      ( ( zero_zero_nat = X3 )
    <=> ( X3 = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_130_zero__reorient,axiom,
    ! [X3: int] :
      ( ( zero_zero_int = X3 )
    <=> ( X3 = zero_zero_int ) ) ).

% zero_reorient
tff(fact_131_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_132_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.assoc
tff(fact_133_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_134_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_135_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).

% mult.assoc
tff(fact_136_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) ).

% mult.assoc
tff(fact_137_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_138_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_139_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_140_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_141_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_142_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_143_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_144_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_145_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_146_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: 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),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_147_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_148_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_149_Suc__inject,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X3) = aa_nat_nat(suc,Y) )
     => ( X3 = Y ) ) ).

% Suc_inject
tff(fact_150_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_151_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_152_mult__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

% mult_0
tff(fact_153_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_154_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_155_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_156_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_157_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_158_nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_induct
tff(fact_159_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X),zero_zero_nat))
     => ( ! [Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y3)))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X)),aa_nat_nat(suc,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_160_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_161_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_162_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_163_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_164_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M2: nat] : N = aa_nat_nat(suc,M2) ) ).

% not0_implies_Suc
tff(fact_165_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(T),X3) = aa_nat_option_nat(some_nat,Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(aa_VEBT_VEBT_set_nat(vEBT_set_vebt,T)),X3),Sx)) ) ) ).

% succ_correct
tff(fact_166_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_167_list__decode_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ~ ! [N2: nat] : X3 != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_168_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,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ( ( X3 != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : X3 != 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_169_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_1: nat] : pp(aa_nat_bool(P,X_1))
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(P,N2))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_170_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% deg_not_0
tff(fact_171_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_172_option_Osize_I4_J,axiom,
    ! [X2: nat] : size_size_option_nat(aa_nat_option_nat(some_nat,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_173_option_Osize_I4_J,axiom,
    ! [X2: num] : size_size_option_num(aa_num_option_num(some_num,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_174_Leaf__0__not,axiom,
    ! [A: bool,B: bool] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_175_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X22: bool,Y21: bool,Y22: bool] :
      ( ( aa_bool_VEBT_VEBT(vEBT_Leaf(X21),X22) = aa_bool_VEBT_VEBT(vEBT_Leaf(Y21),Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X22)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_176_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_177_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_eq
tff(fact_178_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_179_lessI,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).

% lessI
tff(fact_180_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_181_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% neq0_conv
tff(fact_182_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_183_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).

% zero_less_Suc
tff(fact_184_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_185_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_186_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% mult_less_cancel2
tff(fact_187_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_188_linorder__neqE__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ) ).

% linorder_neqE_nat
tff(fact_189_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(P,N2))
         => ? [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
              & ~ pp(aa_nat_bool(P,M3)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_190_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
             => pp(aa_nat_bool(P,M3)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_191_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_irrefl_nat
tff(fact_192_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S2),T))
     => ( S2 != T ) ) ).

% less_not_refl3
tff(fact_193_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_194_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_not_refl
tff(fact_195_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% nat_neq_iff
tff(fact_196_linorder__neqE__linordered__idom,axiom,
    ! [X3: real,Y: real] :
      ( ( X3 != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_197_linorder__neqE__linordered__idom,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_198_linorder__neqE__linordered__idom,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_199_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_200_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_201_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_202_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_203_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_204_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_205_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_206_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_207_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_208_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_209_size__neq__size__imp__neq,axiom,
    ! [X3: list_real,Y: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,X3) != aa_list_real_nat(size_size_list_real,Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_210_size__neq__size__imp__neq,axiom,
    ! [X3: list_bool,Y: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,X3) != aa_list_bool_nat(size_size_list_bool,Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_211_size__neq__size__imp__neq,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,X3) != aa_list_nat_nat(size_size_list_nat,Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_212_size__neq__size__imp__neq,axiom,
    ! [X3: list_int,Y: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,X3) != aa_list_int_nat(size_size_list_int,Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_213_size__neq__size__imp__neq,axiom,
    ! [X3: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X3) != aa_num_nat(size_size_num,Y) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_214_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: bool,X222: bool] : Y != aa_bool_VEBT_VEBT(vEBT_Leaf(X212),X222) ) ).

% VEBT.exhaust
tff(fact_215_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: bool,X22: bool] : aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) != aa_bool_VEBT_VEBT(vEBT_Leaf(X21),X22) ).

% VEBT.distinct(1)
tff(fact_216_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_217_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_218_not__less__zero,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less_zero
tff(fact_219_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr_zeroI
tff(fact_220_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_221_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => pp(aa_nat_bool(P,I2)) )
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I2)))
               => pp(aa_nat_bool(P,I2)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_222_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),aa_nat_nat(suc,I2)))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),K2)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_223_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_224_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_SucD
tff(fact_225_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_226_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
    <=> ? [M4: nat] :
          ( ( M = aa_nat_nat(suc,M4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M4)) ) ) ).

% Suc_less_eq2
tff(fact_227_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
           => pp(aa_nat_bool(P,I3)) ) ) ) ).

% Nat.All_less_Suc
tff(fact_228_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M))) ) ).

% not_less_eq
tff(fact_229_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_230_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
            & pp(aa_nat_bool(P,I3)) ) ) ) ).

% Ex_less_Suc
tff(fact_231_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_232_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_233_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( ( aa_nat_nat(suc,M) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_234_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
     => ~ ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_235_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_lessD
tff(fact_236_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_237_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( ~ pp(aa_nat_bool(P,N2))
             => ? [M3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
                  & ~ pp(aa_nat_bool(P,M3)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_238_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_239_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_zeroE
tff(fact_240_not__less0,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less0
tff(fact_241_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_242_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr0I
tff(fact_243_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_244_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_245_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_246_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_247_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_248_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_249_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_250_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_251_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_252_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_253_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_254_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_255_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_256_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_257_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_258_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_259_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_260_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_261_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_262_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_263_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_264_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_265_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_266_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_267_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_268_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_269_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_270_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_271_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_272_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_273_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_274_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_275_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_276_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_277_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_278_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_279_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_280_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos
tff(fact_281_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_282_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_283_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos
tff(fact_284_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_mult_iff
tff(fact_285_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_mult_iff
tff(fact_286_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_287_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_pos_neg2
tff(fact_288_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_289_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_290_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_291_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_pos_pos
tff(fact_292_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_pos_pos
tff(fact_293_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).

% mult_pos_pos
tff(fact_294_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_pos_pos
tff(fact_295_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_pos_neg
tff(fact_296_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_297_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_298_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_299_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_neg_pos
tff(fact_300_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_301_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_302_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_303_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_304_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_305_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_306_not__square__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real)) ).

% not_square_less_zero
tff(fact_307_not__square__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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_308_not__square__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_309_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_neg_neg
tff(fact_310_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_neg_neg
tff(fact_311_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_neg_neg
tff(fact_312_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_313_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M2: nat] : N = aa_nat_nat(suc,M2) ) ).

% gr0_implies_Suc
tff(fact_314_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% All_less_Suc2
tff(fact_315_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M5: nat] : N = aa_nat_nat(suc,M5) ) ).

% gr0_conv_Suc
tff(fact_316_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% Ex_less_Suc2
tff(fact_317_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_318_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ) ).

% mult_less_mono1
tff(fact_319_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ) ).

% mult_less_mono2
tff(fact_320_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B: bool] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_321_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_322_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_323_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% one_less_mult
tff(fact_324_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(T),X3) = aa_nat_option_nat(some_nat,Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),X3),Sx)) ) ) ).

% succ_corr
tff(fact_325_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( aa_VEBT_VEBT_set_nat(vEBT_set_vebt,T) = aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_326_less__option__Some,axiom,
    ! [X3: real,Y: real] :
      ( ord_less_option_real(aa_real_option_real(some_real,X3),aa_real_option_real(some_real,Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ).

% less_option_Some
tff(fact_327_less__option__Some,axiom,
    ! [X3: rat,Y: rat] :
      ( ord_less_option_rat(some_rat(X3),some_rat(Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y)) ) ).

% less_option_Some
tff(fact_328_less__option__Some,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,aa_num_option_num(some_num,X3)),aa_num_option_num(some_num,Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y)) ) ).

% less_option_Some
tff(fact_329_less__option__Some,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,aa_nat_option_nat(some_nat,X3)),aa_nat_option_nat(some_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y)) ) ).

% less_option_Some
tff(fact_330_less__option__Some,axiom,
    ! [X3: int,Y: int] :
      ( ord_less_option_int(aa_int_option_int(some_int,X3),aa_int_option_int(some_int,Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y)) ) ).

% less_option_Some
tff(fact_331_greater__shift,axiom,
    ! [Y: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,X3)),aa_nat_option_nat(some_nat,Y))) ) ).

% greater_shift
tff(fact_332_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N)),N)) ) ).

% buildup_gives_valid
tff(fact_333_T__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% T_vebt_buildupi_gq_0
tff(fact_334_less__shift,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,X3)),aa_nat_option_nat(some_nat,Y))) ) ).

% less_shift
tff(fact_335_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X3: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,X3)))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),X3))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_336_forall__finite_I2_J,axiom,
    ! [P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,zero_zero_nat)))
         => pp(aa_nat_bool(P,I3)) )
    <=> pp(aa_nat_bool(P,zero_zero_nat)) ) ).

% forall_finite(2)
tff(fact_337_forall__finite_I3_J,axiom,
    ! [X3: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,aa_nat_nat(suc,X3))))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,X3)))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% forall_finite(3)
tff(fact_338_mult__less__iff1,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ) ).

% mult_less_iff1
tff(fact_339_mult__less__iff1,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y)) ) ) ).

% mult_less_iff1
tff(fact_340_mult__less__iff1,axiom,
    ! [Z: int,X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y)) ) ) ).

% mult_less_iff1
tff(fact_341_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X3))
      <=> pp(aa_set_nat_bool(member_nat(X3),aa_VEBT_VEBT_set_nat(vEBT_set_vebt,T))) ) ) ).

% member_correct
tff(fact_342_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T)) ).

% set_vebt'_def
tff(fact_343_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X3: nat] : hoare_2774773592644289642e_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_V854960066525838166emberi(T,Ti,X3),aa_nat_fun_bool_assn(aa_VEB5211276637198020136l_assn(aTP_Lamm_au(T),Ti),X3)) ).

% vebt_memberi'_rf_abstr
tff(fact_344_vebt__buildup_Osimps_I1_J,axiom,
    aa_nat_VEBT_VEBT(vEBT_vebt_buildup,zero_zero_nat) = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) ).

% vebt_buildup.simps(1)
tff(fact_345_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_346_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_347_vebt__buildup_Osimps_I2_J,axiom,
    aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,zero_zero_nat)) = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) ).

% vebt_buildup.simps(2)
tff(fact_348_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X22: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,aa_bool_VEBT_VEBT(vEBT_Leaf(X21),X22)) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_349_forall__finite_I1_J,axiom,
    ! [P: fun_nat_bool,I4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),zero_zero_nat))
     => pp(aa_nat_bool(P,I4)) ) ).

% forall_finite(1)
tff(fact_350_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),X3)) ).

% vebt_member.simps(4)
tff(fact_351_succ__member,axiom,
    ! [T: vEBT_VEBT,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),X3),Y))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
        & ! [Z2: nat] :
            ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Z2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2)) ) ) ) ).

% succ_member
tff(fact_352_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),zero_zero_nat,Uy),Uz)),X3)) ).

% vebt_member.simps(3)
tff(fact_353_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X3: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N)),X3)) ).

% buildup_nothing_in_leaf
tff(fact_354_mul__shift,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,X3)),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,Z) ) ) ).

% mul_shift
tff(fact_355_buildup__gives__empty,axiom,
    ! [N: nat] : aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_356_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,T) = aa_nat_option_nat(some_nat,Maxi) )
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Maxi)) ) ) ).

% maxt_member
tff(fact_357_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X3: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N)),X3)) ).

% buildup_nothing_in_min_max
tff(fact_358_max__in__set__def,axiom,
    ! [Xs: set_nat,X3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(Xs),X3))
    <=> ( pp(aa_set_nat_bool(member_nat(X3),Xs))
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),X3)) ) ) ) ).

% max_in_set_def
tff(fact_359_min__in__set__def,axiom,
    ! [Xs: set_nat,X3: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(Xs),X3))
    <=> ( pp(aa_set_nat_bool(member_nat(X3),Xs))
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),X4)) ) ) ) ).

% min_in_set_def
tff(fact_360_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X3))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,Tree),X3))
          | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,Tree),X3)) ) ) ) ).

% member_valid_both_member_options
tff(fact_361_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,T) = aa_nat_option_nat(some_nat,Maxi) )
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Maxi)) ) ) ) ).

% maxt_corr_help
tff(fact_362_le__zero__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_363_less__eq__option__Some,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ord_le353528952715127954et_int(some_set_int(X3),some_set_int(Y))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y)) ) ).

% less_eq_option_Some
tff(fact_364_less__eq__option__Some,axiom,
    ! [X3: rat,Y: rat] :
      ( ord_le2406147912482264968on_rat(some_rat(X3),some_rat(Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y)) ) ).

% less_eq_option_Some
tff(fact_365_less__eq__option__Some,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,aa_num_option_num(some_num,X3)),aa_num_option_num(some_num,Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y)) ) ).

% less_eq_option_Some
tff(fact_366_less__eq__option__Some,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,aa_nat_option_nat(some_nat,X3)),aa_nat_option_nat(some_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ).

% less_eq_option_Some
tff(fact_367_less__eq__option__Some,axiom,
    ! [X3: int,Y: int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,X3),aa_int_option_int(some_int,Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y)) ) ).

% less_eq_option_Some
tff(fact_368_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% Suc_le_mono
tff(fact_369_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_370_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A)) ).

% bot_nat_0.extremum
tff(fact_371_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,T) = aa_nat_option_nat(some_nat,X3) )
       => pp(aa_nat_bool(vEBT_VEBT_max_in_set(aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),X3)) ) ) ).

% maxt_corr
tff(fact_372_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),X3))
       => ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,T) = aa_nat_option_nat(some_nat,X3) ) ) ) ).

% maxt_sound
tff(fact_373_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).

% one_le_mult_iff
tff(fact_374_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_375_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% mult_le_cancel2
tff(fact_376_lesseq__shift,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,aa_nat_option_nat(some_nat,X3)),aa_nat_option_nat(some_nat,Y))) ) ).

% lesseq_shift
tff(fact_377_pred__member,axiom,
    ! [T: vEBT_VEBT,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),X3),Y))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3))
        & ! [Z2: nat] :
            ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Z2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),Y)) ) ) ) ).

% pred_member
tff(fact_378_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_379_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => ? [X: nat] :
            ( pp(aa_nat_bool(P,X))
            & ! [Y4: nat] :
                ( pp(aa_nat_bool(P,Y4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),X)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_380_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% nat_le_linear
tff(fact_381_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_382_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% eq_imp_le
tff(fact_383_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).

% le_trans
tff(fact_384_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_385_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_386_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_387_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_388_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_389_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_390_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N2))),aa_nat_set_int(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,N3)),aa_nat_set_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_391_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_392_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_393_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_394_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_395_zero__le,axiom,
    ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X3)) ).

% zero_le
tff(fact_396_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X),X))
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X),Z3)) ) )
         => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N2),aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_397_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(P,M))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_398_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M3)),N2))
             => pp(aa_nat_bool(P,M3)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% full_nat_induct
tff(fact_399_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_400_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_401_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        | ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_402_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M6))
     => ? [M2: nat] : M6 = aa_nat_nat(suc,M2) ) ).

% Suc_le_D
tff(fact_403_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N))) ) ).

% le_SucI
tff(fact_404_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_405_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_leD
tff(fact_406_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_407_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_408_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_409_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% less_eq_nat.simps(1)
tff(fact_410_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_411_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ( M != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% le_neq_implies_less
tff(fact_412_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_or_eq_imp_le
tff(fact_413_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_414_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_imp_le_nat
tff(fact_415_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_416_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ).

% mult_le_mono2
tff(fact_417_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ).

% mult_le_mono1
tff(fact_418_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L))) ) ) ).

% mult_le_mono
tff(fact_419_le__square,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_square
tff(fact_420_le__cube,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)))) ).

% le_cube
tff(fact_421_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),Uv)),Uw)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_422_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_423_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_424_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_425_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_426_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_mult_iff
tff(fact_427_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_mult_iff
tff(fact_428_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_429_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_430_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_431_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_432_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_433_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonpos_nonneg
tff(fact_434_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_435_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_436_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_437_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos
tff(fact_438_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_439_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_440_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_441_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_442_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_443_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_444_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_445_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ).

% split_mult_neg_le
tff(fact_446_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_447_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_448_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(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_449_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_450_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_451_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_452_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono
tff(fact_453_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).

% mult_right_mono
tff(fact_454_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_right_mono
tff(fact_455_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono
tff(fact_456_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_457_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_458_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_459_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono
tff(fact_460_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% mult_left_mono
tff(fact_461_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_left_mono
tff(fact_462_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono
tff(fact_463_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_464_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_465_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_466_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_467_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_468_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_469_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ).

% split_mult_pos_le
tff(fact_470_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ).

% split_mult_pos_le
tff(fact_471_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).

% split_mult_pos_le
tff(fact_472_zero__le__square,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A))) ).

% zero_le_square
tff(fact_473_zero__le__square,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A))) ).

% zero_le_square
tff(fact_474_zero__le__square,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A))) ).

% zero_le_square
tff(fact_475_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_476_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_477_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_478_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_479_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_480_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_481_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_482_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_483_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N)) ) ).

% Suc_leI
tff(fact_484_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_eq
tff(fact_485_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,I))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_486_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
                 => pp(aa_nat_bool(P,N2)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_487_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_lessD
tff(fact_488_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_489_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_Suc_eq_le
tff(fact_490_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_491_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_492_ex__least__nat__le,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_493_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_494_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_495_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_496_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_497_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_498_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_499_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_500_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_501_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_502_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_503_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_504_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_505_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_506_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_507_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_508_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_509_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_510_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_511_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_512_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_513_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_514_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_515_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_516_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_517_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_518_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_519_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_520_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_521_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_522_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_523_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_524_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_525_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_526_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_527_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_528_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_529_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_530_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_531_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_532_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_533_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_534_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_535_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_536_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_537_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_538_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_539_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_540_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_541_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_542_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_543_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_544_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_545_mult__le__cancel__iff2,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_546_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_547_mult__le__cancel__iff2,axiom,
    ! [Z: int,X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_548_mult__le__cancel__iff1,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_549_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_550_mult__le__cancel__iff1,axiom,
    ! [Z: int,X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_551_ex__least__nat__less,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),K2))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_552_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_553_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_maxti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_av(T),Ti)) ).

% vebt_maxti_h
tff(fact_554_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(T),X3) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aw(T),X3)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_555_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,T) = none_nat )
       => ( aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_556_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),X3),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X4)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_557_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(suc,A))) )
    <=> ( B = A ) ) ).

% nat_in_between_eq(2)
tff(fact_558_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(suc,A))) )
    <=> ( B = aa_nat_nat(suc,A) ) ) ).

% nat_in_between_eq(1)
tff(fact_559_nat__compl__induct_H,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N2))
               => pp(aa_nat_bool(P,Nn)) )
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_compl_induct'
tff(fact_560_nat__compl__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N2))
               => pp(aa_nat_bool(P,Nn)) )
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_compl_induct
tff(fact_561_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,T) = aa_nat_option_nat(some_nat,Mini) )
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mini),X3)) ) ) ) ).

% mint_corr_help
tff(fact_562_empty__iff,axiom,
    ! [C: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C),bot_bo8194388402131092736T_VEBT)) ).

% empty_iff
tff(fact_563_empty__iff,axiom,
    ! [C: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(C),bot_bot_set_set_nat)) ).

% empty_iff
tff(fact_564_empty__iff,axiom,
    ! [C: real] : ~ pp(aa_set_real_bool(member_real(C),bot_bot_set_real)) ).

% empty_iff
tff(fact_565_empty__iff,axiom,
    ! [C: bool] : ~ pp(aa_set_bool_bool(member_bool(C),bot_bot_set_bool)) ).

% empty_iff
tff(fact_566_empty__iff,axiom,
    ! [C: nat] : ~ pp(aa_set_nat_bool(member_nat(C),bot_bot_set_nat)) ).

% empty_iff
tff(fact_567_empty__iff,axiom,
    ! [C: int] : ~ pp(aa_set_int_bool(member_int(C),bot_bot_set_int)) ).

% empty_iff
tff(fact_568_all__not__in__conv,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A2))
    <=> ( A2 = bot_bo8194388402131092736T_VEBT ) ) ).

% all_not_in_conv
tff(fact_569_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_570_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] : ~ pp(aa_set_real_bool(member_real(X4),A2))
    <=> ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_571_all__not__in__conv,axiom,
    ! [A2: set_bool] :
      ( ! [X4: bool] : ~ pp(aa_set_bool_bool(member_bool(X4),A2))
    <=> ( A2 = bot_bot_set_bool ) ) ).

% all_not_in_conv
tff(fact_572_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ~ pp(aa_set_nat_bool(member_nat(X4),A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_573_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ~ pp(aa_set_int_bool(member_int(X4),A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_574_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,T) = aa_nat_option_nat(some_nat,Maxi) )
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Maxi)) ) ) ).

% mint_member
tff(fact_575_empty__subsetI,axiom,
    ! [A2: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A2)) ).

% empty_subsetI
tff(fact_576_empty__subsetI,axiom,
    ! [A2: set_bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(bot_bot_set_bool),A2)) ).

% empty_subsetI
tff(fact_577_empty__subsetI,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A2)) ).

% empty_subsetI
tff(fact_578_empty__subsetI,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A2)) ).

% empty_subsetI
tff(fact_579_subset__empty,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_580_subset__empty,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),bot_bot_set_bool))
    <=> ( A2 = bot_bot_set_bool ) ) ).

% subset_empty
tff(fact_581_subset__empty,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_582_subset__empty,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_583_empty__Collect__eq,axiom,
    ! [P: fun_complex_bool] :
      ( ( bot_bot_set_complex = collect_complex(P) )
    <=> ! [X4: complex] : ~ pp(aa_complex_bool(P,X4)) ) ).

% empty_Collect_eq
tff(fact_584_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X4: list_nat] : ~ pp(aa_list_nat_bool(P,X4)) ) ).

% empty_Collect_eq
tff(fact_585_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X4: set_nat] : ~ pp(aa_set_nat_bool(P,X4)) ) ).

% empty_Collect_eq
tff(fact_586_empty__Collect__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X4: real] : ~ pp(aa_real_bool(P,X4)) ) ).

% empty_Collect_eq
tff(fact_587_empty__Collect__eq,axiom,
    ! [P: fun_bool_bool] :
      ( ( bot_bot_set_bool = collect_bool(P) )
    <=> ! [X4: bool] : ~ pp(aa_bool_bool(P,X4)) ) ).

% empty_Collect_eq
tff(fact_588_empty__Collect__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X4: nat] : ~ pp(aa_nat_bool(P,X4)) ) ).

% empty_Collect_eq
tff(fact_589_empty__Collect__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X4: int] : ~ pp(aa_int_bool(P,X4)) ) ).

% empty_Collect_eq
tff(fact_590_Collect__empty__eq,axiom,
    ! [P: fun_complex_bool] :
      ( ( collect_complex(P) = bot_bot_set_complex )
    <=> ! [X4: complex] : ~ pp(aa_complex_bool(P,X4)) ) ).

% Collect_empty_eq
tff(fact_591_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X4: list_nat] : ~ pp(aa_list_nat_bool(P,X4)) ) ).

% Collect_empty_eq
tff(fact_592_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] : ~ pp(aa_set_nat_bool(P,X4)) ) ).

% Collect_empty_eq
tff(fact_593_Collect__empty__eq,axiom,
    ! [P: fun_real_bool] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X4: real] : ~ pp(aa_real_bool(P,X4)) ) ).

% Collect_empty_eq
tff(fact_594_Collect__empty__eq,axiom,
    ! [P: fun_bool_bool] :
      ( ( collect_bool(P) = bot_bot_set_bool )
    <=> ! [X4: bool] : ~ pp(aa_bool_bool(P,X4)) ) ).

% Collect_empty_eq
tff(fact_595_Collect__empty__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X4: nat] : ~ pp(aa_nat_bool(P,X4)) ) ).

% Collect_empty_eq
tff(fact_596_Collect__empty__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X4: int] : ~ pp(aa_int_bool(P,X4)) ) ).

% Collect_empty_eq
tff(fact_597_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,T) = none_nat )
       => ( aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_598_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,T) = aa_nat_option_nat(some_nat,X3) )
       => pp(aa_nat_bool(vEBT_VEBT_min_in_set(aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),X3)) ) ) ).

% mint_corr
tff(fact_599_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),X3))
       => ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,T) = aa_nat_option_nat(some_nat,X3) ) ) ) ).

% mint_sound
tff(fact_600_not__None__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ( X3 != none_P5556105721700978146at_nat )
    <=> ? [Y5: product_prod_nat_nat] : X3 = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Y5) ) ).

% not_None_eq
tff(fact_601_not__None__eq,axiom,
    ! [X3: option_nat] :
      ( ( X3 != none_nat )
    <=> ? [Y5: nat] : X3 = aa_nat_option_nat(some_nat,Y5) ) ).

% not_None_eq
tff(fact_602_not__None__eq,axiom,
    ! [X3: option_num] :
      ( ( X3 != none_num )
    <=> ? [Y5: num] : X3 = aa_num_option_num(some_num,Y5) ) ).

% not_None_eq
tff(fact_603_not__Some__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ! [Y5: product_prod_nat_nat] : X3 != aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Y5)
    <=> ( X3 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_604_not__Some__eq,axiom,
    ! [X3: option_nat] :
      ( ! [Y5: nat] : X3 != aa_nat_option_nat(some_nat,Y5)
    <=> ( X3 = none_nat ) ) ).

% not_Some_eq
tff(fact_605_not__Some__eq,axiom,
    ! [X3: option_num] :
      ( ! [Y5: num] : X3 != aa_num_option_num(some_num,Y5)
    <=> ( X3 = none_num ) ) ).

% not_Some_eq
tff(fact_606_less__eq__option__None__code,axiom,
    ! [X3: option_nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,none_nat),X3)) ).

% less_eq_option_None_code
tff(fact_607_less__eq__option__None__code,axiom,
    ! [X3: option_num] : pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,none_num),X3)) ).

% less_eq_option_None_code
tff(fact_608_less__option__None,axiom,
    ! [X3: option_nat] : ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,X3),none_nat)) ).

% less_option_None
tff(fact_609_less__option__None,axiom,
    ! [X3: option_num] : ~ pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,X3),none_num)) ).

% less_option_None
tff(fact_610_less__eq__option__Some__None,axiom,
    ! [X3: nat] : ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,aa_nat_option_nat(some_nat,X3)),none_nat)) ).

% less_eq_option_Some_None
tff(fact_611_less__eq__option__Some__None,axiom,
    ! [X3: num] : ~ pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,aa_num_option_num(some_num,X3)),none_num)) ).

% less_eq_option_Some_None
tff(fact_612_less__option__None__Some__code,axiom,
    ! [X3: nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,none_nat),aa_nat_option_nat(some_nat,X3))) ).

% less_option_None_Some_code
tff(fact_613_less__option__None__Some__code,axiom,
    ! [X3: num] : pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,none_num),aa_num_option_num(some_num,X3))) ).

% less_option_None_Some_code
tff(fact_614_less__eq__option__None__is__None,axiom,
    ! [X3: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,X3),none_nat))
     => ( X3 = none_nat ) ) ).

% less_eq_option_None_is_None
tff(fact_615_less__eq__option__None__is__None,axiom,
    ! [X3: option_num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,X3),none_num))
     => ( X3 = none_num ) ) ).

% less_eq_option_None_is_None
tff(fact_616_less__eq__option__None,axiom,
    ! [X3: option_nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,none_nat),X3)) ).

% less_eq_option_None
tff(fact_617_less__eq__option__None,axiom,
    ! [X3: option_num] : pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,none_num),X3)) ).

% less_eq_option_None
tff(fact_618_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X2) ).

% option.distinct(1)
tff(fact_619_option_Odistinct_I1_J,axiom,
    ! [X2: nat] : none_nat != aa_nat_option_nat(some_nat,X2) ).

% option.distinct(1)
tff(fact_620_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != aa_num_option_num(some_num,X2) ).

% option.distinct(1)
tff(fact_621_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_622_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option = aa_nat_option_nat(some_nat,X2) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_623_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = aa_num_option_num(some_num,X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_624_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X23) ) ).

% option.exhaust
tff(fact_625_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != aa_nat_option_nat(some_nat,X23) ) ).

% option.exhaust
tff(fact_626_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != aa_num_option_num(some_num,X23) ) ).

% option.exhaust
tff(fact_627_split__option__ex,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ? [X_12: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_12))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        | ? [X4: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X4))) ) ) ).

% split_option_ex
tff(fact_628_split__option__ex,axiom,
    ! [P: fun_option_nat_bool] :
      ( ? [X_12: option_nat] : pp(aa_option_nat_bool(P,X_12))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        | ? [X4: nat] : pp(aa_option_nat_bool(P,aa_nat_option_nat(some_nat,X4))) ) ) ).

% split_option_ex
tff(fact_629_split__option__ex,axiom,
    ! [P: fun_option_num_bool] :
      ( ? [X_12: option_num] : pp(aa_option_num_bool(P,X_12))
    <=> ( pp(aa_option_num_bool(P,none_num))
        | ? [X4: num] : pp(aa_option_num_bool(P,aa_num_option_num(some_num,X4))) ) ) ).

% split_option_ex
tff(fact_630_split__option__all,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ! [X_12: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_12))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        & ! [X4: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X4))) ) ) ).

% split_option_all
tff(fact_631_split__option__all,axiom,
    ! [P: fun_option_nat_bool] :
      ( ! [X_12: option_nat] : pp(aa_option_nat_bool(P,X_12))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        & ! [X4: nat] : pp(aa_option_nat_bool(P,aa_nat_option_nat(some_nat,X4))) ) ) ).

% split_option_all
tff(fact_632_split__option__all,axiom,
    ! [P: fun_option_num_bool] :
      ( ! [X_12: option_num] : pp(aa_option_num_bool(P,X_12))
    <=> ( pp(aa_option_num_bool(P,none_num))
        & ! [X4: num] : pp(aa_option_num_bool(P,aa_num_option_num(some_num,X4))) ) ) ).

% split_option_all
tff(fact_633_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: fun_op5682630191948829869t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X3),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X3),Y)) )
       => ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( X3 = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,A3) )
             => ( ( Y = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B2) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X3),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X3),Y)) ) ) ) ).

% combine_options_cases
tff(fact_634_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: fun_op9100744243108201484t_bool,Y: option_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X3),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X3),Y)) )
       => ( ! [A3: product_prod_nat_nat,B2: nat] :
              ( ( X3 = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,A3) )
             => ( ( Y = aa_nat_option_nat(some_nat,B2) )
               => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X3),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X3),Y)) ) ) ) ).

% combine_options_cases
tff(fact_635_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: fun_op775672374849087042m_bool,Y: option_num] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X3),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X3),Y)) )
       => ( ! [A3: product_prod_nat_nat,B2: num] :
              ( ( X3 = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,A3) )
             => ( ( Y = aa_num_option_num(some_num,B2) )
               => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X3),Y)) ) )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X3),Y)) ) ) ) ).

% combine_options_cases
tff(fact_636_combine__options__cases,axiom,
    ! [X3: option_nat,P: fun_op5698422665285572208t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X3 = none_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X3),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X3),Y)) )
       => ( ! [A3: nat,B2: product_prod_nat_nat] :
              ( ( X3 = aa_nat_option_nat(some_nat,A3) )
             => ( ( Y = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B2) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X3),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X3),Y)) ) ) ) ).

% combine_options_cases
tff(fact_637_combine__options__cases,axiom,
    ! [X3: option_nat,P: fun_op7792301606653096457t_bool,Y: option_nat] :
      ( ( ( X3 = none_nat )
       => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X3),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X3),Y)) )
       => ( ! [A3: nat,B2: nat] :
              ( ( X3 = aa_nat_option_nat(some_nat,A3) )
             => ( ( Y = aa_nat_option_nat(some_nat,B2) )
               => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X3),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X3),Y)) ) ) ) ).

% combine_options_cases
tff(fact_638_combine__options__cases,axiom,
    ! [X3: option_nat,P: fun_op8690601775248757823m_bool,Y: option_num] :
      ( ( ( X3 = none_nat )
       => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X3),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X3),Y)) )
       => ( ! [A3: nat,B2: num] :
              ( ( X3 = aa_nat_option_nat(some_nat,A3) )
             => ( ( Y = aa_num_option_num(some_num,B2) )
               => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X3),Y)) ) )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X3),Y)) ) ) ) ).

% combine_options_cases
tff(fact_639_combine__options__cases,axiom,
    ! [X3: option_num,P: fun_op7732454363416259258t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X3 = none_num )
       => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X3),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X3),Y)) )
       => ( ! [A3: num,B2: product_prod_nat_nat] :
              ( ( X3 = aa_num_option_num(some_num,A3) )
             => ( ( Y = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B2) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X3),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X3),Y)) ) ) ) ).

% combine_options_cases
tff(fact_640_combine__options__cases,axiom,
    ! [X3: option_num,P: fun_op1064277265382982207t_bool,Y: option_nat] :
      ( ( ( X3 = none_num )
       => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X3),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X3),Y)) )
       => ( ! [A3: num,B2: nat] :
              ( ( X3 = aa_num_option_num(some_num,A3) )
             => ( ( Y = aa_nat_option_nat(some_nat,B2) )
               => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X3),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X3),Y)) ) ) ) ).

% combine_options_cases
tff(fact_641_combine__options__cases,axiom,
    ! [X3: option_num,P: fun_op1962577433978643573m_bool,Y: option_num] :
      ( ( ( X3 = none_num )
       => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X3),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X3),Y)) )
       => ( ! [A3: num,B2: num] :
              ( ( X3 = aa_num_option_num(some_num,A3) )
             => ( ( Y = aa_num_option_num(some_num,B2) )
               => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X3),Y)) ) )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X3),Y)) ) ) ) ).

% combine_options_cases
tff(fact_642_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv),Uw)),aa_nat_nat(suc,N)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_643_less__option__None__is__Some,axiom,
    ! [X3: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,none_nat),X3))
     => ? [Z3: nat] : X3 = aa_nat_option_nat(some_nat,Z3) ) ).

% less_option_None_is_Some
tff(fact_644_less__option__None__is__Some,axiom,
    ! [X3: option_num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,none_num),X3))
     => ? [Z3: num] : X3 = aa_num_option_num(some_num,Z3) ) ).

% less_option_None_is_Some
tff(fact_645_less__option__None__Some,axiom,
    ! [X3: nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,none_nat),aa_nat_option_nat(some_nat,X3))) ).

% less_option_None_Some
tff(fact_646_less__option__None__Some,axiom,
    ! [X3: num] : pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,none_num),aa_num_option_num(some_num,X3))) ).

% less_option_None_Some
tff(fact_647_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),zero_zero_nat,Vc),Vd)),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_648_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( A = B )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C))
       => ( ( C = D )
         => pp(aa_set_int_bool(ord_less_eq_set_int(A),D)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_649_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ( C = D )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),D)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_650_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ( C = D )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),D)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_651_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ( C = D )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),D)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_652_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => ( ( C = D )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),D)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_653_memb__imp__not__empty,axiom,
    ! [X3: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),S3))
     => ( S3 != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
tff(fact_654_memb__imp__not__empty,axiom,
    ! [X3: set_nat,S3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),S3))
     => ( S3 != bot_bot_set_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_655_memb__imp__not__empty,axiom,
    ! [X3: real,S3: set_real] :
      ( pp(aa_set_real_bool(member_real(X3),S3))
     => ( S3 != bot_bot_set_real ) ) ).

% memb_imp_not_empty
tff(fact_656_memb__imp__not__empty,axiom,
    ! [X3: bool,S3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(X3),S3))
     => ( S3 != bot_bot_set_bool ) ) ).

% memb_imp_not_empty
tff(fact_657_memb__imp__not__empty,axiom,
    ! [X3: nat,S3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),S3))
     => ( S3 != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_658_memb__imp__not__empty,axiom,
    ! [X3: int,S3: set_int] :
      ( pp(aa_set_int_bool(member_int(X3),S3))
     => ( S3 != bot_bot_set_int ) ) ).

% memb_imp_not_empty
tff(fact_659_set__notEmptyE,axiom,
    ! [S3: set_VEBT_VEBT] :
      ( ( S3 != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3)) ) ).

% set_notEmptyE
tff(fact_660_set__notEmptyE,axiom,
    ! [S3: set_set_nat] :
      ( ( S3 != bot_bot_set_set_nat )
     => ~ ! [X: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X),S3)) ) ).

% set_notEmptyE
tff(fact_661_set__notEmptyE,axiom,
    ! [S3: set_real] :
      ( ( S3 != bot_bot_set_real )
     => ~ ! [X: real] : ~ pp(aa_set_real_bool(member_real(X),S3)) ) ).

% set_notEmptyE
tff(fact_662_set__notEmptyE,axiom,
    ! [S3: set_bool] :
      ( ( S3 != bot_bot_set_bool )
     => ~ ! [X: bool] : ~ pp(aa_set_bool_bool(member_bool(X),S3)) ) ).

% set_notEmptyE
tff(fact_663_set__notEmptyE,axiom,
    ! [S3: set_nat] :
      ( ( S3 != bot_bot_set_nat )
     => ~ ! [X: nat] : ~ pp(aa_set_nat_bool(member_nat(X),S3)) ) ).

% set_notEmptyE
tff(fact_664_set__notEmptyE,axiom,
    ! [S3: set_int] :
      ( ( S3 != bot_bot_set_int )
     => ~ ! [X: int] : ~ pp(aa_set_int_bool(member_int(X),S3)) ) ).

% set_notEmptyE
tff(fact_665_ex__in__conv,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ? [X4: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A2))
    <=> ( A2 != bot_bo8194388402131092736T_VEBT ) ) ).

% ex_in_conv
tff(fact_666_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ? [X4: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
    <=> ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_667_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ? [X4: real] : pp(aa_set_real_bool(member_real(X4),A2))
    <=> ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_668_ex__in__conv,axiom,
    ! [A2: set_bool] :
      ( ? [X4: bool] : pp(aa_set_bool_bool(member_bool(X4),A2))
    <=> ( A2 != bot_bot_set_bool ) ) ).

% ex_in_conv
tff(fact_669_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X4: nat] : pp(aa_set_nat_bool(member_nat(X4),A2))
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_670_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X4: int] : pp(aa_set_int_bool(member_int(X4),A2))
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_671_equals0I,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ! [Y3: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y3),A2))
     => ( A2 = bot_bo8194388402131092736T_VEBT ) ) ).

% equals0I
tff(fact_672_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y3: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(Y3),A2))
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_673_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y3: real] : ~ pp(aa_set_real_bool(member_real(Y3),A2))
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_674_equals0I,axiom,
    ! [A2: set_bool] :
      ( ! [Y3: bool] : ~ pp(aa_set_bool_bool(member_bool(Y3),A2))
     => ( A2 = bot_bot_set_bool ) ) ).

% equals0I
tff(fact_675_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y3: nat] : ~ pp(aa_set_nat_bool(member_nat(Y3),A2))
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_676_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y3: int] : ~ pp(aa_set_int_bool(member_int(Y3),A2))
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_677_equals0D,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( A2 = bot_bo8194388402131092736T_VEBT )
     => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),A2)) ) ).

% equals0D
tff(fact_678_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ pp(aa_set_set_nat_bool(member_set_nat(A),A2)) ) ).

% equals0D
tff(fact_679_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ pp(aa_set_real_bool(member_real(A),A2)) ) ).

% equals0D
tff(fact_680_equals0D,axiom,
    ! [A2: set_bool,A: bool] :
      ( ( A2 = bot_bot_set_bool )
     => ~ pp(aa_set_bool_bool(member_bool(A),A2)) ) ).

% equals0D
tff(fact_681_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ pp(aa_set_nat_bool(member_nat(A),A2)) ) ).

% equals0D
tff(fact_682_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ pp(aa_set_int_bool(member_int(A),A2)) ) ).

% equals0D
tff(fact_683_emptyE,axiom,
    ! [A: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)) ).

% emptyE
tff(fact_684_emptyE,axiom,
    ! [A: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(A),bot_bot_set_set_nat)) ).

% emptyE
tff(fact_685_emptyE,axiom,
    ! [A: real] : ~ pp(aa_set_real_bool(member_real(A),bot_bot_set_real)) ).

% emptyE
tff(fact_686_emptyE,axiom,
    ! [A: bool] : ~ pp(aa_set_bool_bool(member_bool(A),bot_bot_set_bool)) ).

% emptyE
tff(fact_687_emptyE,axiom,
    ! [A: nat] : ~ pp(aa_set_nat_bool(member_nat(A),bot_bot_set_nat)) ).

% emptyE
tff(fact_688_emptyE,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(member_int(A),bot_bot_set_int)) ).

% emptyE
tff(fact_689_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A2),bot_bot_set_real)) ).

% not_psubset_empty
tff(fact_690_not__psubset__empty,axiom,
    ! [A2: set_bool] : ~ pp(aa_set_bool_bool(ord_less_set_bool(A2),bot_bot_set_bool)) ).

% not_psubset_empty
tff(fact_691_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),bot_bot_set_nat)) ).

% not_psubset_empty
tff(fact_692_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A2),bot_bot_set_int)) ).

% not_psubset_empty
tff(fact_693_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_694_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_695_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_696_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_697_Set_Oempty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_ax) ).

% Set.empty_def
tff(fact_698_Set_Oempty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_ay) ).

% Set.empty_def
tff(fact_699_Set_Oempty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_az) ).

% Set.empty_def
tff(fact_700_Set_Oempty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_ba) ).

% Set.empty_def
tff(fact_701_Set_Oempty__def,axiom,
    bot_bot_set_bool = collect_bool(aTP_Lamm_bb) ).

% Set.empty_def
tff(fact_702_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_bc) ).

% Set.empty_def
tff(fact_703_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_bd) ).

% Set.empty_def
tff(fact_704_le__some__optE,axiom,
    ! [M: set_int,X3: option_set_int] :
      ( ord_le353528952715127954et_int(some_set_int(M),X3)
     => ~ ! [M7: set_int] :
            ( ( X3 = some_set_int(M7) )
           => ~ pp(aa_set_int_bool(ord_less_eq_set_int(M),M7)) ) ) ).

% le_some_optE
tff(fact_705_le__some__optE,axiom,
    ! [M: rat,X3: option_rat] :
      ( ord_le2406147912482264968on_rat(some_rat(M),X3)
     => ~ ! [M7: rat] :
            ( ( X3 = some_rat(M7) )
           => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,M),M7)) ) ) ).

% le_some_optE
tff(fact_706_le__some__optE,axiom,
    ! [M: num,X3: option_num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,aa_num_option_num(some_num,M)),X3))
     => ~ ! [M7: num] :
            ( ( X3 = aa_num_option_num(some_num,M7) )
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),M7)) ) ) ).

% le_some_optE
tff(fact_707_le__some__optE,axiom,
    ! [M: nat,X3: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,aa_nat_option_nat(some_nat,M)),X3))
     => ~ ! [M7: nat] :
            ( ( X3 = aa_nat_option_nat(some_nat,M7) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),M7)) ) ) ).

% le_some_optE
tff(fact_708_le__some__optE,axiom,
    ! [M: int,X3: option_int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,M),X3)
     => ~ ! [M7: int] :
            ( ( X3 = aa_int_option_int(some_int,M7) )
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),M7)) ) ) ).

% le_some_optE
tff(fact_709_exists__leI,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ( ! [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N4),N))
           => ~ pp(aa_nat_bool(P,N4)) )
       => pp(aa_nat_bool(P,N)) )
     => ? [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N))
          & pp(aa_nat_bool(P,N5)) ) ) ).

% exists_leI
tff(fact_710_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_minti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_be(T),Ti)) ).

% vebt_minti_h
tff(fact_711_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(T),X3) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bf(T),X3)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_712_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(T),X3) = aa_nat_option_nat(some_nat,Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(aa_VEBT_VEBT_set_nat(vEBT_set_vebt,T)),X3),Sx)) ) ) ).

% pred_correct
tff(fact_713_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Px: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(T),X3) = aa_nat_option_nat(some_nat,Px) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),X3),Px)) ) ) ).

% pred_corr
tff(fact_714_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Ma),X3))
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_715_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),X3),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3))
        & ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),X3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_716_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_717_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw,aa_num_option_num(some_num,V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_718_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),aa_nat_option_nat(some_nat,V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_719_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: fun_Pr2477239776880934807at_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X3,Xa,Xb) = Y )
     => ( ( ( Xa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xa = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2)
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,A3) )
               => ! [B2: product_prod_nat_nat] :
                    ( ( Xb = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B2) )
                   => ( Y != aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X3,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_720_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: fun_num_fun_num_num,Xa: option_num,Xb: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X3,Xa,Xb) = Y )
     => ( ( ( Xa = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xa = aa_num_option_num(some_num,V2)
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xa = aa_num_option_num(some_num,A3) )
               => ! [B2: num] :
                    ( ( Xb = aa_num_option_num(some_num,B2) )
                   => ( Y != aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(X3,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_721_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: fun_nat_fun_nat_nat,Xa: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X3),Xa),Xb) = Y )
     => ( ( ( Xa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xa = aa_nat_option_nat(some_nat,V2)
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa = aa_nat_option_nat(some_nat,A3) )
               => ! [B2: nat] :
                    ( ( Xb = aa_nat_option_nat(some_nat,B2) )
                   => ( Y != aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(X3,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_722_obtain__set__pred,axiom,
    ! [Z: nat,X3: nat,A2: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X3))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(A2),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
         => ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(A2),X3),X_13)) ) ) ) ).

% obtain_set_pred
tff(fact_723_obtain__set__succ,axiom,
    ! [X3: nat,Z: nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(A2),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
         => ( ( A2 = B3 )
           => ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(A2),X3),X_13)) ) ) ) ) ).

% obtain_set_succ
tff(fact_724_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T))) ) ).

% set_vebt_finite
tff(fact_725_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),A),X_13))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),X5)) ) ) ) ).

% succ_none_empty
tff(fact_726_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),A),X_13))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),A)) ) ) ) ).

% pred_none_empty
tff(fact_727_subsetI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set_nat_bool(member_nat(X),B3)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3)) ) ).

% subsetI
tff(fact_728_subsetI,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),B3)) )
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3)) ) ).

% subsetI
tff(fact_729_subsetI,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set_real_bool(member_real(X),B3)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3)) ) ).

% subsetI
tff(fact_730_subsetI,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ! [X: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(X),B3)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3)) ) ).

% subsetI
tff(fact_731_subsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_set_int_bool(member_int(X),B3)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ).

% subsetI
tff(fact_732_subset__antisym,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
       => ( A2 = B3 ) ) ) ).

% subset_antisym
tff(fact_733_not__Some__eq2,axiom,
    ! [V: option8994051521412888707BT_nat] :
      ( ! [X4: vEBT_VEBT,Y5: nat] : V != some_P3304306041812697742BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X4),Y5))
    <=> ( V = none_P2898468544434266898BT_nat ) ) ).

% not_Some_eq2
tff(fact_734_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ! [X4: int,Y5: int] : V != some_P4184893108420464158nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X4),Y5))
    <=> ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
tff(fact_735_not__Some__eq2,axiom,
    ! [V: option2651255830984564193nteger] :
      ( ! [X4: code_integer,Y5: code_integer] : V != some_P6772290148444788224nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X4),Y5))
    <=> ( V = none_P4506660739021792380nteger ) ) ).

% not_Some_eq2
tff(fact_736_not__Some__eq2,axiom,
    ! [V: option642762832853965969at_num] :
      ( ! [X4: nat,Y5: num] : V != some_P8071634352977444016at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X4),Y5))
    <=> ( V = none_P6264349658649815852at_num ) ) ).

% not_Some_eq2
tff(fact_737_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ! [X4: nat,Y5: nat] : V != aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y5))
    <=> ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
tff(fact_738_psubsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( ( A2 != B3 )
       => pp(aa_set_int_bool(ord_less_set_int(A2),B3)) ) ) ).

% psubsetI
tff(fact_739_psubsetE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
         => pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ) ).

% psubsetE
tff(fact_740_psubset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
        & ( A2 != B3 ) ) ) ).

% psubset_eq
tff(fact_741_psubset__imp__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ).

% psubset_imp_subset
tff(fact_742_psubset__subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),C2))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C2)) ) ) ).

% psubset_subset_trans
tff(fact_743_subset__not__subset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ) ).

% subset_not_subset_eq
tff(fact_744_subset__psubset__trans,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_set_int(B3),C2))
       => pp(aa_set_int_bool(ord_less_set_int(A2),C2)) ) ) ).

% subset_psubset_trans
tff(fact_745_subset__iff__psubset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A2),B3))
        | ( A2 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_746_in__mono,axiom,
    ! [A2: set_nat,B3: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( pp(aa_set_nat_bool(member_nat(X3),A2))
       => pp(aa_set_nat_bool(member_nat(X3),B3)) ) ) ).

% in_mono
tff(fact_747_in__mono,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),B3)) ) ) ).

% in_mono
tff(fact_748_in__mono,axiom,
    ! [A2: set_real,B3: set_real,X3: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
     => ( pp(aa_set_real_bool(member_real(X3),A2))
       => pp(aa_set_real_bool(member_real(X3),B3)) ) ) ).

% in_mono
tff(fact_749_in__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,X3: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(X3),B3)) ) ) ).

% in_mono
tff(fact_750_in__mono,axiom,
    ! [A2: set_int,B3: set_int,X3: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(member_int(X3),A2))
       => pp(aa_set_int_bool(member_int(X3),B3)) ) ) ).

% in_mono
tff(fact_751_subsetD,axiom,
    ! [A2: set_nat,B3: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( pp(aa_set_nat_bool(member_nat(C),A2))
       => pp(aa_set_nat_bool(member_nat(C),B3)) ) ) ).

% subsetD
tff(fact_752_subsetD,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C),A2))
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(C),B3)) ) ) ).

% subsetD
tff(fact_753_subsetD,axiom,
    ! [A2: set_real,B3: set_real,C: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
     => ( pp(aa_set_real_bool(member_real(C),A2))
       => pp(aa_set_real_bool(member_real(C),B3)) ) ) ).

% subsetD
tff(fact_754_subsetD,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
     => ( pp(aa_set_set_nat_bool(member_set_nat(C),A2))
       => pp(aa_set_set_nat_bool(member_set_nat(C),B3)) ) ) ).

% subsetD
tff(fact_755_subsetD,axiom,
    ! [A2: set_int,B3: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(member_int(C),A2))
       => pp(aa_set_int_bool(member_int(C),B3)) ) ) ).

% subsetD
tff(fact_756_equalityE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ) ).

% equalityE
tff(fact_757_subset__eq,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_nat_bool(member_nat(X4),B3)) ) ) ).

% subset_eq
tff(fact_758_subset__eq,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
    <=> ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),A2))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),B3)) ) ) ).

% subset_eq
tff(fact_759_subset__eq,axiom,
    ! [A2: set_real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
    <=> ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),A2))
         => pp(aa_set_real_bool(member_real(X4),B3)) ) ) ).

% subset_eq
tff(fact_760_subset__eq,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
    <=> ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(X4),B3)) ) ) ).

% subset_eq
tff(fact_761_subset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
    <=> ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_set_int_bool(member_int(X4),B3)) ) ) ).

% subset_eq
tff(fact_762_equalityD1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ).

% equalityD1
tff(fact_763_Set_OequalityD2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ).

% Set.equalityD2
tff(fact_764_subset__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
    <=> ! [T2: nat] :
          ( pp(aa_set_nat_bool(member_nat(T2),A2))
         => pp(aa_set_nat_bool(member_nat(T2),B3)) ) ) ).

% subset_iff
tff(fact_765_subset__iff,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
    <=> ! [T2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(T2),A2))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(T2),B3)) ) ) ).

% subset_iff
tff(fact_766_subset__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
    <=> ! [T2: real] :
          ( pp(aa_set_real_bool(member_real(T2),A2))
         => pp(aa_set_real_bool(member_real(T2),B3)) ) ) ).

% subset_iff
tff(fact_767_subset__iff,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
    <=> ! [T2: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(T2),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(T2),B3)) ) ) ).

% subset_iff
tff(fact_768_subset__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
    <=> ! [T2: int] :
          ( pp(aa_set_int_bool(member_int(T2),A2))
         => pp(aa_set_int_bool(member_int(T2),B3)) ) ) ).

% subset_iff
tff(fact_769_subset__refl,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),A2)) ).

% subset_refl
tff(fact_770_Collect__mono,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ! [X: complex] :
          ( pp(aa_complex_bool(P,X))
         => pp(aa_complex_bool(Q,X)) )
     => pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q))) ) ).

% Collect_mono
tff(fact_771_Collect__mono,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X: list_nat] :
          ( pp(aa_list_nat_bool(P,X))
         => pp(aa_list_nat_bool(Q,X)) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_772_Collect__mono,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X: set_nat] :
          ( pp(aa_set_nat_bool(P,X))
         => pp(aa_set_nat_bool(Q,X)) )
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q))) ) ).

% Collect_mono
tff(fact_773_Collect__mono,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X: nat] :
          ( pp(aa_nat_bool(P,X))
         => pp(aa_nat_bool(Q,X)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q))) ) ).

% Collect_mono
tff(fact_774_Collect__mono,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X: int] :
          ( pp(aa_int_bool(P,X))
         => pp(aa_int_bool(Q,X)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q))) ) ).

% Collect_mono
tff(fact_775_subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),C2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A2),C2)) ) ) ).

% subset_trans
tff(fact_776_set__eq__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ) ).

% set_eq_subset
tff(fact_777_Collect__subset,axiom,
    ! [A2: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bg(A2),P))),A2)) ).

% Collect_subset
tff(fact_778_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bh(A2),P))),A2)) ).

% Collect_subset
tff(fact_779_Collect__subset,axiom,
    ! [A2: set_complex,P: fun_complex_bool] : pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_bi(A2),P))),A2)) ).

% Collect_subset
tff(fact_780_Collect__subset,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bj(A2),P)),A2) ).

% Collect_subset
tff(fact_781_Collect__subset,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bk(A2),P))),A2)) ).

% Collect_subset
tff(fact_782_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bl(A2),P))),A2)) ).

% Collect_subset
tff(fact_783_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bm(A2),P))),A2)) ).

% Collect_subset
tff(fact_784_less__eq__set__def,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
    <=> ord_le8840051076831633728t_bool(aa_set3744802623701456604t_bool(aTP_Lamm_as,A2),aa_set3744802623701456604t_bool(aTP_Lamm_as,B3)) ) ).

% less_eq_set_def
tff(fact_785_less__eq__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
    <=> ord_le7893757273149057542T_bool(aa_set1728193350978918526T_bool(aTP_Lamm_an,A2),aa_set1728193350978918526T_bool(aTP_Lamm_an,B3)) ) ).

% less_eq_set_def
tff(fact_786_less__eq__set__def,axiom,
    ! [A2: set_real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
    <=> ord_le5573776112191778788l_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ao,A2),aa_set8565328311858951076l_bool(aTP_Lamm_ao,B3)) ) ).

% less_eq_set_def
tff(fact_787_less__eq__set__def,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
    <=> ord_le1461730252015488778t_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ar,A2),aa_set9121396911425987952t_bool(aTP_Lamm_ar,B3)) ) ).

% less_eq_set_def
tff(fact_788_less__eq__set__def,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
    <=> ord_le5715760110363009508t_bool(aa_set7012339369581096484t_bool(aTP_Lamm_at,A2),aa_set7012339369581096484t_bool(aTP_Lamm_at,B3)) ) ).

% less_eq_set_def
tff(fact_789_Collect__mono__iff,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q)))
    <=> ! [X4: complex] :
          ( pp(aa_complex_bool(P,X4))
         => pp(aa_complex_bool(Q,X4)) ) ) ).

% Collect_mono_iff
tff(fact_790_Collect__mono__iff,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q))
    <=> ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P,X4))
         => pp(aa_list_nat_bool(Q,X4)) ) ) ).

% Collect_mono_iff
tff(fact_791_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)))
    <=> ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P,X4))
         => pp(aa_set_nat_bool(Q,X4)) ) ) ).

% Collect_mono_iff
tff(fact_792_Collect__mono__iff,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q)))
    <=> ! [X4: nat] :
          ( pp(aa_nat_bool(P,X4))
         => pp(aa_nat_bool(Q,X4)) ) ) ).

% Collect_mono_iff
tff(fact_793_Collect__mono__iff,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(P)),collect_int(Q)))
    <=> ! [X4: int] :
          ( pp(aa_int_bool(P,X4))
         => pp(aa_int_bool(Q,X4)) ) ) ).

% Collect_mono_iff
tff(fact_794_subset__Collect__conv,axiom,
    ! [S3: set_complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),collect_complex(P)))
    <=> ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),S3))
         => pp(aa_complex_bool(P,X4)) ) ) ).

% subset_Collect_conv
tff(fact_795_subset__Collect__conv,axiom,
    ! [S3: set_list_nat,P: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(S3,collect_list_nat(P))
    <=> ! [X4: list_nat] :
          ( pp(aa_set_list_nat_bool(member_list_nat(X4),S3))
         => pp(aa_list_nat_bool(P,X4)) ) ) ).

% subset_Collect_conv
tff(fact_796_subset__Collect__conv,axiom,
    ! [S3: set_set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(S3),collect_set_nat(P)))
    <=> ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),S3))
         => pp(aa_set_nat_bool(P,X4)) ) ) ).

% subset_Collect_conv
tff(fact_797_subset__Collect__conv,axiom,
    ! [S3: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),collect_nat(P)))
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),S3))
         => pp(aa_nat_bool(P,X4)) ) ) ).

% subset_Collect_conv
tff(fact_798_subset__Collect__conv,axiom,
    ! [S3: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),collect_int(P)))
    <=> ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),S3))
         => pp(aa_int_bool(P,X4)) ) ) ).

% subset_Collect_conv
tff(fact_799_prod__decode__aux_Ocases,axiom,
    ! [X3: product_prod_nat_nat] :
      ~ ! [K2: nat,M2: nat] : X3 != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M2) ).

% prod_decode_aux.cases
tff(fact_800_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va2)),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_801_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_802_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_803_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),X3)) ).

% vebt_member.simps(2)
tff(fact_804_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : aa_nat_option_nat(vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),Uv)),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_805_filter__preserves__multiset,axiom,
    ! [M8: fun_list_nat_nat,P: fun_list_nat_bool] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_bn(M8)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bo(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_806_filter__preserves__multiset,axiom,
    ! [M8: fun_set_nat_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_bp(M8))))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bq(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_807_filter__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_br(M8))))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bs(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_808_filter__preserves__multiset,axiom,
    ! [M8: fun_int_nat,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_bt(M8))))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bu(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_809_filter__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_bv(M8))))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_bw(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_810_filter__preserves__multiset,axiom,
    ! [M8: fun_Code_integer_nat,P: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_bx(M8))))
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_by(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_811_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = aa_nat_option_nat(some_nat,Mi) ).

% vebt_mint.simps(3)
tff(fact_812_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = aa_nat_option_nat(some_nat,Ma) ).

% vebt_maxt.simps(3)
tff(fact_813_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),zero_zero_nat,Vd),Ve)),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_814_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_815_add__mset__in__multiset,axiom,
    ! [M8: fun_list_nat_nat,A: list_nat] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_bn(M8)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_bz(M8),A))) ) ).

% add_mset_in_multiset
tff(fact_816_add__mset__in__multiset,axiom,
    ! [M8: fun_set_nat_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_bp(M8))))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_ca(M8),A)))) ) ).

% add_mset_in_multiset
tff(fact_817_add__mset__in__multiset,axiom,
    ! [M8: fun_nat_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_br(M8))))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cb(M8),A)))) ) ).

% add_mset_in_multiset
tff(fact_818_add__mset__in__multiset,axiom,
    ! [M8: fun_int_nat,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_bt(M8))))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_cc(M8),A)))) ) ).

% add_mset_in_multiset
tff(fact_819_add__mset__in__multiset,axiom,
    ! [M8: fun_complex_nat,A: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_bv(M8))))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_cd(M8),A)))) ) ).

% add_mset_in_multiset
tff(fact_820_add__mset__in__multiset,axiom,
    ! [M8: fun_Code_integer_nat,A: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_bx(M8))))
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_ce(M8),A)))) ) ).

% add_mset_in_multiset
tff(fact_821_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va2) = none_nat ).

% vebt_succ.simps(3)
tff(fact_822_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_823_vebt__pred_Osimps_I2_J,axiom,
    ! [A: bool,Uw: bool] :
      ( ( pp(A)
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A),Uw)),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
      & ( ~ pp(A)
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A),Uw)),aa_nat_nat(suc,zero_zero_nat)) = none_nat ) ) ) ).

% vebt_pred.simps(2)
tff(fact_824_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2),Vb)),X3))
    <=> ( ( X3 = Mi )
        | ( X3 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_825_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,A),aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B)) = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_826_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A: num,B: num] : vEBT_V819420779217536731ft_num(F,aa_num_option_num(some_num,A),aa_num_option_num(some_num,B)) = aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_827_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),aa_nat_option_nat(some_nat,A)),aa_nat_option_nat(some_nat,B)) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_828_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cf,K)))) ).

% finite_Collect_le_nat
tff(fact_829_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,K)))) ).

% finite_Collect_less_nat
tff(fact_830_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_ch,A2)))) ) ).

% finite_Collect_subsets
tff(fact_831_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_ci(A2))) ) ).

% finite_Collect_subsets
tff(fact_832_finite__Collect__subsets,axiom,
    ! [A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => finite6931041176100689706nteger(collec574505750873337192nteger(aTP_Lamm_cj(A2))) ) ).

% finite_Collect_subsets
tff(fact_833_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite6197958912794628473et_int(collect_set_int(aa_set7694327796961020398t_bool(aTP_Lamm_ck,A2))) ) ).

% finite_Collect_subsets
tff(fact_834_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ! [A3: bool,B2: bool] : X3 != aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_835_finite__Collect__conjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_cl(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_836_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        | pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) )
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_cm(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_837_finite__Collect__conjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        | pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) )
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_cn(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_838_finite__Collect__conjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        | pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_co(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_839_finite__Collect__conjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        | pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) )
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_cp(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_840_finite__Collect__conjI,axiom,
    ! [P: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool] :
      ( ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(P)))
        | pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(Q))) )
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_cq(P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_841_finite__Collect__disjI,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_cr(P),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_842_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_cs(P),Q))))
    <=> ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
        & pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_843_finite__Collect__disjI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ct(P),Q))))
    <=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
        & pp(aa_set_nat_bool(finite_finite_nat,collect_nat(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_844_finite__Collect__disjI,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_cu(P),Q))))
    <=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
        & pp(aa_set_int_bool(finite_finite_int,collect_int(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_845_finite__Collect__disjI,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_cv(P),Q))))
    <=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_846_finite__Collect__disjI,axiom,
    ! [P: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_cw(P),Q))))
    <=> ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(P)))
        & pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(Q))) ) ) ).

% finite_Collect_disjI
tff(fact_847_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,
    ! [X3: vEBT_VEBT] :
      ( ( X3 != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
     => ( ! [Uv2: bool] : X3 != aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2)
       => ( ! [Uu2: bool] : X3 != aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_848_infinite__growing,axiom,
    ! [X6: set_Code_integer] :
      ( ( X6 != bot_bo3990330152332043303nteger )
     => ( ! [X: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),X6))
           => ? [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),X6))
                & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Xa2)) ) )
       => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,X6)) ) ) ).

% infinite_growing
tff(fact_849_infinite__growing,axiom,
    ! [X6: set_bool] :
      ( ( X6 != bot_bot_set_bool )
     => ( ! [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),X6))
           => ? [Xa2: bool] :
                ( pp(aa_set_bool_bool(member_bool(Xa2),X6))
                & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X),Xa2)) ) )
       => ~ pp(aa_set_bool_bool(finite_finite_bool,X6)) ) ) ).

% infinite_growing
tff(fact_850_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),X6))
           => ? [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),X6))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Xa2)) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,X6)) ) ) ).

% infinite_growing
tff(fact_851_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X: rat] :
            ( pp(aa_set_rat_bool(member_rat(X),X6))
           => ? [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),X6))
                & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Xa2)) ) )
       => ~ pp(aa_set_rat_bool(finite_finite_rat,X6)) ) ) ).

% infinite_growing
tff(fact_852_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X: num] :
            ( pp(aa_set_num_bool(member_num(X),X6))
           => ? [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),X6))
                & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Xa2)) ) )
       => ~ pp(aa_set_num_bool(finite_finite_num,X6)) ) ) ).

% infinite_growing
tff(fact_853_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),X6))
           => ? [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),X6))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Xa2)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X6)) ) ) ).

% infinite_growing
tff(fact_854_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),X6))
           => ? [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),X6))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa2)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X6)) ) ) ).

% infinite_growing
tff(fact_855_ex__min__if__finite,axiom,
    ! [S3: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ? [X: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),S3))
            & ~ ? [Xa2: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),S3))
                  & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Xa2),X)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_856_ex__min__if__finite,axiom,
    ! [S3: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ( ( S3 != bot_bot_set_bool )
       => ? [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),S3))
            & ~ ? [Xa2: bool] :
                  ( pp(aa_set_bool_bool(member_bool(Xa2),S3))
                  & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,Xa2),X)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_857_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( S3 != bot_bot_set_real )
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),S3))
            & ~ ? [Xa2: real] :
                  ( pp(aa_set_real_bool(member_real(Xa2),S3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa2),X)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_858_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,S3))
     => ( ( S3 != bot_bot_set_rat )
       => ? [X: rat] :
            ( pp(aa_set_rat_bool(member_rat(X),S3))
            & ~ ? [Xa2: rat] :
                  ( pp(aa_set_rat_bool(member_rat(Xa2),S3))
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa2),X)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_859_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,S3))
     => ( ( S3 != bot_bot_set_num )
       => ? [X: num] :
            ( pp(aa_set_num_bool(member_num(X),S3))
            & ~ ? [Xa2: num] :
                  ( pp(aa_set_num_bool(member_num(Xa2),S3))
                  & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Xa2),X)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_860_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),S3))
            & ~ ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),S3))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),X)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_861_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( S3 != bot_bot_set_int )
       => ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),S3))
            & ~ ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),S3))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa2),X)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_862_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : X3 != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : X3 != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] : X3 != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,A3),aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_863_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : X3 != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : X3 != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B2: nat] : X3 != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,A3),aa_nat_option_nat(some_nat,B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_864_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : X3 != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_num,V2: num] : X3 != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(aa_num_option_num(some_num,V2),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A3: num,B2: num] : X3 != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(aa_num_option_num(some_num,A3),aa_num_option_num(some_num,B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_865_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc533602882674032673at_nat] :
      ( ! [Uu2: fun_Pr248683899324676269t_bool,Uv2: option4927543243414619207at_nat] : X3 != produc5961576194287291153at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr248683899324676269t_bool,V2: product_prod_nat_nat] : X3 != produc5961576194287291153at_nat(Uw2,produc488173922507101015at_nat(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr248683899324676269t_bool,X: product_prod_nat_nat,Y3: product_prod_nat_nat] : X3 != produc5961576194287291153at_nat(F2,produc488173922507101015at_nat(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X),aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_866_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc3700678068921409119on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_bool,Uv2: option_nat] : X3 != produc2004789812759622991on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_bool,V2: nat] : X3 != produc2004789812759622991on_nat(Uw2,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_bool,X: nat,Y3: nat] : X3 != produc2004789812759622991on_nat(F2,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,X),aa_nat_option_nat(some_nat,Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_867_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc6167219322466932359on_num] :
      ( ! [Uu2: fun_num_fun_num_bool,Uv2: option_num] : X3 != produc8433281988922931063on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_bool,V2: num] : X3 != produc8433281988922931063on_num(Uw2,produc8585076106096196333on_num(aa_num_option_num(some_num,V2),none_num))
       => ~ ! [F2: fun_num_fun_num_bool,X: num,Y3: num] : X3 != produc8433281988922931063on_num(F2,produc8585076106096196333on_num(aa_num_option_num(some_num,X),aa_num_option_num(some_num,Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_868_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,D2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)),D2)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_869_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B2: bool,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),X)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S)),X) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_870_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)),zero_zero_nat)
     => ( ! [A3: bool,Uw2: bool] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: bool,B2: bool,Va: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2)),Vf2)
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_871_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,B2: bool] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2)),zero_zero_nat)
     => ( ! [Uv2: bool,Uw2: bool,N2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)),aa_nat_nat(suc,N2))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Va3)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2)),Ve2)
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_872_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,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B2: bool,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),X)
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S)),X)
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S)),X)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2)),X)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_873_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,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B2: bool,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),X)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),X)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2)),X)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),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] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_874_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_nat,R: fun_VE3763956218935105199t_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
             => ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),B3))
                  & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(R,X),Xa2)) ) )
         => ? [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),B3))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_cx(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_875_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_nat,R: fun_re9121486324101951821t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),A2))
             => ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),B3))
                  & pp(aa_nat_bool(aa_real_fun_nat_bool(R,X),Xa2)) ) )
         => ? [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),B3))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_cy(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_876_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_int,R: fun_VE3303673267288458579t_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B3))
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
             => ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),B3))
                  & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(R,X),Xa2)) ) )
         => ? [X: int] :
              ( pp(aa_set_int_bool(member_int(X),B3))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_cz(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_877_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_int,R: fun_re8661203372455305201t_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B3))
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),A2))
             => ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),B3))
                  & pp(aa_int_bool(aa_real_fun_int_bool(R,X),Xa2)) ) )
         => ? [X: int] :
              ( pp(aa_set_int_bool(member_int(X),B3))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_da(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_878_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_complex,R: fun_VE2128610149277289041x_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
             => ? [Xa2: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa2),B3))
                  & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(R,X),Xa2)) ) )
         => ? [X: complex] :
              ( pp(aa_set_complex_bool(member_complex(X),B3))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_db(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_879_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_complex,R: fun_re3010880487650781423x_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),A2))
             => ? [Xa2: complex] :
                  ( pp(aa_set_complex_bool(member_complex(Xa2),B3))
                  & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(R,X),Xa2)) ) )
         => ? [X: complex] :
              ( pp(aa_set_complex_bool(member_complex(X),B3))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_dc(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_880_pigeonhole__infinite__rel,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_Code_integer,R: fun_VE1123255499321568226r_bool] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
             => ? [Xa2: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),B3))
                  & pp(aa_Code_integer_bool(aa_VEB6607917098650726363r_bool(R,X),Xa2)) ) )
         => ? [X: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),B3))
              & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_Cod3526525079615536833T_bool(aa_fun7592865282689444384T_bool(aTP_Lamm_dd(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_881_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_Code_integer,R: fun_re1120004649140997760r_bool] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),A2))
             => ? [Xa2: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),B3))
                  & pp(aa_Code_integer_bool(aa_rea7928672538722535017r_bool(R,X),Xa2)) ) )
         => ? [X: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),B3))
              & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_Cod181991726960451049l_bool(aa_fun8452463775060891964l_bool(aTP_Lamm_de(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_882_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B3: set_nat,R: fun_nat_fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( ! [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),A2))
             => ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),B3))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X),Xa2)) ) )
         => ? [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),B3))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_df(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_883_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B3: set_int,R: fun_nat_fun_int_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B3))
       => ( ! [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),A2))
             => ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),B3))
                  & pp(aa_int_bool(aa_nat_fun_int_bool(R,X),Xa2)) ) )
         => ? [X: int] :
              ( pp(aa_set_int_bool(member_int(X),B3))
              & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_dg(A2),R),X)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_884_not__finite__existsD,axiom,
    ! [P: fun_list_nat_bool] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P))
     => ? [X_13: list_nat] : pp(aa_list_nat_bool(P,X_13)) ) ).

% not_finite_existsD
tff(fact_885_not__finite__existsD,axiom,
    ! [P: fun_set_nat_bool] :
      ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(P)))
     => ? [X_13: set_nat] : pp(aa_set_nat_bool(P,X_13)) ) ).

% not_finite_existsD
tff(fact_886_not__finite__existsD,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => ? [X_13: nat] : pp(aa_nat_bool(P,X_13)) ) ).

% not_finite_existsD
tff(fact_887_not__finite__existsD,axiom,
    ! [P: fun_int_bool] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,collect_int(P)))
     => ? [X_13: int] : pp(aa_int_bool(P,X_13)) ) ).

% not_finite_existsD
tff(fact_888_not__finite__existsD,axiom,
    ! [P: fun_complex_bool] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(P)))
     => ? [X_13: complex] : pp(aa_complex_bool(P,X_13)) ) ).

% not_finite_existsD
tff(fact_889_not__finite__existsD,axiom,
    ! [P: fun_Co8822979462182099027r_bool] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(P)))
     => ? [X_13: code_integer] : pp(aa_Code_integer_bool(P,X_13)) ) ).

% not_finite_existsD
tff(fact_890_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2)),X)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)),X)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)),X) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_891_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_892_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ? [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),A))
            & ! [Xa2: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa2),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_893_finite__has__minimal2,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A),A2))
       => ? [X: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),A2))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),A))
            & ! [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),A2))
               => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_894_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X: set_int] :
            ( member_set_int(X,A2)
            & pp(aa_set_int_bool(ord_less_eq_set_int(X),A))
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_895_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(member_rat(A),A2))
       => ? [X: rat] :
            ( pp(aa_set_rat_bool(member_rat(X),A2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_896_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(member_num(A),A2))
       => ? [X: num] :
            ( pp(aa_set_num_bool(member_num(X),A2))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),A))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_897_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_898_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_899_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_900_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
       => ? [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),X))
            & ! [Xa2: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Xa2),A2))
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_901_finite__has__maximal2,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A),A2))
       => ? [X: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),A2))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),X))
            & ! [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),A2))
               => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_902_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X: set_int] :
            ( member_set_int(X,A2)
            & pp(aa_set_int_bool(ord_less_eq_set_int(A),X))
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_903_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(member_rat(A),A2))
       => ? [X: rat] :
            ( pp(aa_set_rat_bool(member_rat(X),A2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_904_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(member_num(A),A2))
       => ? [X: num] :
            ( pp(aa_set_num_bool(member_num(X),A2))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),X))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_905_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_906_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_907_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_908_infinite__imp__nonempty,axiom,
    ! [S3: set_Code_integer] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( S3 != bot_bo3990330152332043303nteger ) ) ).

% infinite_imp_nonempty
tff(fact_909_infinite__imp__nonempty,axiom,
    ! [S3: set_real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S3))
     => ( S3 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_910_infinite__imp__nonempty,axiom,
    ! [S3: set_bool] :
      ( ~ pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ( S3 != bot_bot_set_bool ) ) ).

% infinite_imp_nonempty
tff(fact_911_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_912_infinite__imp__nonempty,axiom,
    ! [S3: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
     => ( S3 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_913_finite_OemptyI,axiom,
    pp(aa_set_complex_bool(finite3207457112153483333omplex,bot_bot_set_complex)) ).

% finite.emptyI
tff(fact_914_finite_OemptyI,axiom,
    pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,bot_bo3990330152332043303nteger)) ).

% finite.emptyI
tff(fact_915_finite_OemptyI,axiom,
    pp(aa_set_real_bool(finite_finite_real,bot_bot_set_real)) ).

% finite.emptyI
tff(fact_916_finite_OemptyI,axiom,
    pp(aa_set_bool_bool(finite_finite_bool,bot_bot_set_bool)) ).

% finite.emptyI
tff(fact_917_finite_OemptyI,axiom,
    pp(aa_set_nat_bool(finite_finite_nat,bot_bot_set_nat)) ).

% finite.emptyI
tff(fact_918_finite_OemptyI,axiom,
    pp(aa_set_int_bool(finite_finite_int,bot_bot_set_int)) ).

% finite.emptyI
tff(fact_919_rev__finite__subset,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
       => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% rev_finite_subset
tff(fact_920_rev__finite__subset,axiom,
    ! [B3: set_complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B3))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% rev_finite_subset
tff(fact_921_rev__finite__subset,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A2),B3))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2)) ) ) ).

% rev_finite_subset
tff(fact_922_rev__finite__subset,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
       => pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% rev_finite_subset
tff(fact_923_infinite__super,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
       => ~ pp(aa_set_nat_bool(finite_finite_nat,T3)) ) ) ).

% infinite_super
tff(fact_924_infinite__super,axiom,
    ! [S3: set_complex,T3: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T3))
     => ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,T3)) ) ) ).

% infinite_super
tff(fact_925_infinite__super,axiom,
    ! [S3: set_Code_integer,T3: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(S3),T3))
     => ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
       => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,T3)) ) ) ).

% infinite_super
tff(fact_926_infinite__super,axiom,
    ! [S3: set_int,T3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
       => ~ pp(aa_set_int_bool(finite_finite_int,T3)) ) ) ).

% infinite_super
tff(fact_927_finite__subset,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% finite_subset
tff(fact_928_finite__subset,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B3))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% finite_subset
tff(fact_929_finite__subset,axiom,
    ! [A2: set_Code_integer,B3: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A2),B3))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2)) ) ) ).

% finite_subset
tff(fact_930_finite__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(finite_finite_int,B3))
       => pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% finite_subset
tff(fact_931_finite__has__minimal,axiom,
    ! [A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ? [X: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),A2))
            & ! [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),A2))
               => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_932_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_933_finite__has__minimal,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ? [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),A2))
            & ! [Xa2: bool] :
                ( pp(aa_set_bool_bool(member_bool(Xa2),A2))
               => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,Xa2),X))
                 => ( pp(X)
                  <=> pp(Xa2) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_934_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X: set_int] :
            ( member_set_int(X,A2)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_935_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ? [X: rat] :
            ( pp(aa_set_rat_bool(member_rat(X),A2))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_936_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ? [X: num] :
            ( pp(aa_set_num_bool(member_num(X),A2))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_937_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_938_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa2),X))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_939_finite__has__maximal,axiom,
    ! [A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ? [X: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),A2))
            & ! [Xa2: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),A2))
               => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_940_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
            & ! [Xa2: real] :
                ( pp(aa_set_real_bool(member_real(Xa2),A2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_941_finite__has__maximal,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ? [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),A2))
            & ! [Xa2: bool] :
                ( pp(aa_set_bool_bool(member_bool(Xa2),A2))
               => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X),Xa2))
                 => ( pp(X)
                  <=> pp(Xa2) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_942_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X: set_int] :
            ( member_set_int(X,A2)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A2)
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_943_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ? [X: rat] :
            ( pp(aa_set_rat_bool(member_rat(X),A2))
            & ! [Xa2: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa2),A2))
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_944_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ? [X: num] :
            ( pp(aa_set_num_bool(member_num(X),A2))
            & ! [Xa2: num] :
                ( pp(aa_set_num_bool(member_num(Xa2),A2))
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_945_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
            & ! [Xa2: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa2),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_946_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
            & ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_947_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,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B2: bool] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),zero_zero_nat)
     => ( ! [A3: bool,B2: bool] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: bool,B2: bool,N2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,aa_nat_nat(suc,N2)))
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)),Uu2)
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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] : X3 != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_948_vebt__maxt_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,X3) = Y )
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ~ ( ( pp(B2)
                 => ( Y = aa_nat_option_nat(some_nat,one_one_nat) ) )
                & ( ~ pp(B2)
                 => ( ( pp(A3)
                     => ( Y = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
                    & ( ~ pp(A3)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != aa_nat_option_nat(some_nat,Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_949_vebt__mint_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,X3) = Y )
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ~ ( ( pp(A3)
                 => ( Y = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
                & ( ~ pp(A3)
                 => ( ( pp(B2)
                     => ( Y = aa_nat_option_nat(some_nat,one_one_nat) ) )
                    & ( ~ pp(B2)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != aa_nat_option_nat(some_nat,Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_950_pred__subset__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ord_le8840051076831633728t_bool(aa_set3744802623701456604t_bool(aTP_Lamm_as,R),aa_set3744802623701456604t_bool(aTP_Lamm_as,S3))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,R),S3)) ) ).

% pred_subset_eq
tff(fact_951_pred__subset__eq,axiom,
    ! [R: set_VEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ord_le7893757273149057542T_bool(aa_set1728193350978918526T_bool(aTP_Lamm_an,R),aa_set1728193350978918526T_bool(aTP_Lamm_an,S3))
    <=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(R),S3)) ) ).

% pred_subset_eq
tff(fact_952_pred__subset__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ord_le5573776112191778788l_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ao,R),aa_set8565328311858951076l_bool(aTP_Lamm_ao,S3))
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(R),S3)) ) ).

% pred_subset_eq
tff(fact_953_pred__subset__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat] :
      ( ord_le1461730252015488778t_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ar,R),aa_set9121396911425987952t_bool(aTP_Lamm_ar,S3))
    <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(R),S3)) ) ).

% pred_subset_eq
tff(fact_954_pred__subset__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ord_le5715760110363009508t_bool(aa_set7012339369581096484t_bool(aTP_Lamm_at,R),aa_set7012339369581096484t_bool(aTP_Lamm_at,S3))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(R),S3)) ) ).

% pred_subset_eq
tff(fact_955_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(F,N2)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dh(F),U)))) ) ).

% finite_less_ub
tff(fact_956_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_di(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_957_dual__order_Orefl,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% dual_order.refl
tff(fact_958_dual__order_Orefl,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% dual_order.refl
tff(fact_959_dual__order_Orefl,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% dual_order.refl
tff(fact_960_dual__order_Orefl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% dual_order.refl
tff(fact_961_dual__order_Orefl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% dual_order.refl
tff(fact_962_order__refl,axiom,
    ! [X3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X3),X3)) ).

% order_refl
tff(fact_963_order__refl,axiom,
    ! [X3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),X3)) ).

% order_refl
tff(fact_964_order__refl,axiom,
    ! [X3: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),X3)) ).

% order_refl
tff(fact_965_order__refl,axiom,
    ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),X3)) ).

% order_refl
tff(fact_966_order__refl,axiom,
    ! [X3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),X3)) ).

% order_refl
tff(fact_967_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
    <=> ? [A4: bool,B4: bool] : T = aa_bool_VEBT_VEBT(vEBT_Leaf(A4),B4) ) ).

% deg1Leaf
tff(fact_968_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
     => ? [A3: bool,B2: bool] : T = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) ) ).

% deg_1_Leaf
tff(fact_969_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( N = one_one_nat )
       => ? [A3: bool,B2: bool] : T = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) ) ) ).

% deg_1_Leafy
tff(fact_970_star__false__right,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_971_star__false__left,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),P) = bot_bot_assn ).

% star_false_left
tff(fact_972_false__rule,axiom,
    ! [C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] : hoare_7629718768684598413on_nat(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_973_false__rule,axiom,
    ! [C: heap_Time_Heap_bool,Q: fun_bool_assn] : hoare_2774773592644289642e_bool(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_974_false__rule,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] : hoare_1429296392585015714_VEBTi(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_975_false__rule,axiom,
    ! [C: heap_Time_Heap_nat,Q: fun_nat_assn] : hoare_3067605981109127869le_nat(bot_bot_assn,C,Q) ).

% false_rule
tff(fact_976_pure__assn__eq__false__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_assn(pure_assn,P) = bot_bot_assn )
    <=> ~ pp(P) ) ).

% pure_assn_eq_false_iff
tff(fact_977_pure__false,axiom,
    aa_bool_assn(pure_assn,fFalse) = bot_bot_assn ).

% pure_false
tff(fact_978_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_979_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_980_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_981_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_982_mult__1,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% mult_1
tff(fact_983_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,one_one_complex),A) = A ).

% mult_1
tff(fact_984_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_985_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_986_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_987_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_988_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_989_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_990_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_991_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_992_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_993_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_994_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_995_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_996_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_997_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_998_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_999_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_1000_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_1001_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1002_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1003_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_1004_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1005_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1006_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1007_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_1008_less__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_1009_one__reorient,axiom,
    ! [X3: assn] :
      ( ( one_one_assn = X3 )
    <=> ( X3 = one_one_assn ) ) ).

% one_reorient
tff(fact_1010_one__reorient,axiom,
    ! [X3: real] :
      ( ( one_one_real = X3 )
    <=> ( X3 = one_one_real ) ) ).

% one_reorient
tff(fact_1011_one__reorient,axiom,
    ! [X3: rat] :
      ( ( one_one_rat = X3 )
    <=> ( X3 = one_one_rat ) ) ).

% one_reorient
tff(fact_1012_one__reorient,axiom,
    ! [X3: nat] :
      ( ( one_one_nat = X3 )
    <=> ( X3 = one_one_nat ) ) ).

% one_reorient
tff(fact_1013_one__reorient,axiom,
    ! [X3: int] :
      ( ( one_one_int = X3 )
    <=> ( X3 = one_one_int ) ) ).

% one_reorient
tff(fact_1014_bot__empty__eq,axiom,
    ! [X5: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(bot_bo1239524807443452474T_bool,X5))
    <=> pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),bot_bo8194388402131092736T_VEBT)) ) ).

% bot_empty_eq
tff(fact_1015_bot__empty__eq,axiom,
    ! [X5: set_nat] :
      ( pp(aa_set_nat_bool(bot_bo8343409700046576086t_bool,X5))
    <=> pp(aa_set_set_nat_bool(member_set_nat(X5),bot_bot_set_set_nat)) ) ).

% bot_empty_eq
tff(fact_1016_bot__empty__eq,axiom,
    ! [X5: real] :
      ( pp(aa_real_bool(bot_bo4181092155253064880l_bool,X5))
    <=> pp(aa_set_real_bool(member_real(X5),bot_bot_set_real)) ) ).

% bot_empty_eq
tff(fact_1017_bot__empty__eq,axiom,
    ! [X5: bool] :
      ( pp(aa_bool_bool(bot_bo4756488196683355415l_bool,X5))
    <=> pp(aa_set_bool_bool(member_bool(X5),bot_bot_set_bool)) ) ).

% bot_empty_eq
tff(fact_1018_bot__empty__eq,axiom,
    ! [X5: nat] :
      ( pp(aa_nat_bool(bot_bot_fun_nat_bool,X5))
    <=> pp(aa_set_nat_bool(member_nat(X5),bot_bot_set_nat)) ) ).

% bot_empty_eq
tff(fact_1019_bot__empty__eq,axiom,
    ! [X5: int] :
      ( pp(aa_int_bool(bot_bot_fun_int_bool,X5))
    <=> pp(aa_set_int_bool(member_int(X5),bot_bot_set_int)) ) ).

% bot_empty_eq
tff(fact_1020_bot__set__def,axiom,
    bot_bot_set_complex = collect_complex(bot_bo8490223647127498926x_bool) ).

% bot_set_def
tff(fact_1021_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo6469247547558333820t_bool) ).

% bot_set_def
tff(fact_1022_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo8343409700046576086t_bool) ).

% bot_set_def
tff(fact_1023_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bo4181092155253064880l_bool) ).

% bot_set_def
tff(fact_1024_bot__set__def,axiom,
    bot_bot_set_bool = collect_bool(bot_bo4756488196683355415l_bool) ).

% bot_set_def
tff(fact_1025_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_bool) ).

% bot_set_def
tff(fact_1026_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_bool) ).

% bot_set_def
tff(fact_1027_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1028_bot__option__def,axiom,
    bot_bot_option_nat = none_nat ).

% bot_option_def
tff(fact_1029_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
tff(fact_1030_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1031_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1032_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1033_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1034_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1035_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_1036_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_1037_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_1038_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_1039_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_1040_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_1041_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_1042_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_1043_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_1044_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_1045_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_1046_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1047_nat__mult__1__right,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

% nat_mult_1_right
tff(fact_1048_nat__mult__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

% nat_mult_1
tff(fact_1049_bot__empty__eq2,axiom,
    ! [X5: nat,Xa2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bot_bo6560894463593755925t_bool,X5),Xa2))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa2),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_1050_bot__empty__eq2,axiom,
    ! [X5: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(bot_bo8694918001306130243t_bool,X5),Xa2))
    <=> member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X5),Xa2),bot_bo1642239108664514429BT_nat) ) ).

% bot_empty_eq2
tff(fact_1051_bot__empty__eq2,axiom,
    ! [X5: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(bot_bo6207810721336151133t_bool,X5),Xa2))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X5),Xa2),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_1052_bot__empty__eq2,axiom,
    ! [X5: code_integer,Xa2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(bot_bo7217057988103639803r_bool,X5),Xa2))
    <=> member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X5),Xa2),bot_bo4276436098303576167nteger) ) ).

% bot_empty_eq2
tff(fact_1053_bot__empty__eq2,axiom,
    ! [X5: nat,Xa2: num] :
      ( pp(aa_num_bool(aa_nat_fun_num_bool(bot_bo531702463448601163m_bool,X5),Xa2))
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X5),Xa2),bot_bo7038385379056416535at_num) ) ).

% bot_empty_eq2
tff(fact_1054_lambda__one,axiom,
    aTP_Lamm_dj = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_1055_lambda__one,axiom,
    aTP_Lamm_dk = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).

% lambda_one
tff(fact_1056_lambda__one,axiom,
    aTP_Lamm_dl = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_1057_lambda__one,axiom,
    aTP_Lamm_dm = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_1058_lambda__one,axiom,
    aTP_Lamm_dn = aa_ass954072293579886259n_assn(times_times_assn,one_one_assn) ).

% lambda_one
tff(fact_1059_lambda__one,axiom,
    aTP_Lamm_do = aa_com3342990543769683525omplex(times_times_complex,one_one_complex) ).

% lambda_one
tff(fact_1060_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% zero_less_one_class.zero_le_one
tff(fact_1061_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1062_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1063_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% zero_less_one_class.zero_le_one
tff(fact_1064_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1065_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1066_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1067_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1068_not__one__le__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

% not_one_le_zero
tff(fact_1069_not__one__le__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).

% not_one_le_zero
tff(fact_1070_not__one__le__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

% not_one_le_zero
tff(fact_1071_not__one__le__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

% not_one_le_zero
tff(fact_1072_zero__less__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% zero_less_one
tff(fact_1073_zero__less__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one
tff(fact_1074_zero__less__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one
tff(fact_1075_zero__less__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% zero_less_one
tff(fact_1076_not__one__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

% not_one_less_zero
tff(fact_1077_not__one__less__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).

% not_one_less_zero
tff(fact_1078_not__one__less__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

% not_one_less_zero
tff(fact_1079_not__one__less__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).

% not_one_less_zero
tff(fact_1080_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),N))) ) ) ).

% less_1_mult
tff(fact_1081_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),N))) ) ) ).

% less_1_mult
tff(fact_1082_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% less_1_mult
tff(fact_1083_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))) ) ) ).

% less_1_mult
tff(fact_1084_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1085_nat__geq__1__eq__neqz,axiom,
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),X3))
    <=> ( X3 != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_1086_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1087_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),Uv)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_1088_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1089_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1090_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1091_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1092_le__cases3,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
       => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) )
     => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Z)) )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Z))
           => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) )
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y))
             => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3)) )
           => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
               => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X3)) )
             => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X3))
                 => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1093_le__cases3,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
       => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z)) )
     => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Z)) )
       => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Z))
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) )
         => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y))
             => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3)) )
           => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
               => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X3)) )
             => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X3))
                 => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1094_le__cases3,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Z)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Z))
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) )
           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
               => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X3)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X3))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1095_le__cases3,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Z)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Z))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3)) )
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
               => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X3)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X3))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1096_order__class_Oorder__eq__iff,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( X3 = Y )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
        & pp(aa_set_int_bool(ord_less_eq_set_int(Y),X3)) ) ) ).

% order_class.order_eq_iff
tff(fact_1097_order__class_Oorder__eq__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 = Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3)) ) ) ).

% order_class.order_eq_iff
tff(fact_1098_order__class_Oorder__eq__iff,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 = Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3)) ) ) ).

% order_class.order_eq_iff
tff(fact_1099_order__class_Oorder__eq__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ) ).

% order_class.order_eq_iff
tff(fact_1100_order__class_Oorder__eq__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 = Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3)) ) ) ).

% order_class.order_eq_iff
tff(fact_1101_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_1102_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_1103_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_1104_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_1105_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_1106_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( ( B = C )
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_1107_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_1108_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_1109_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_1110_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_1111_order__antisym,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X3))
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_1112_order__antisym,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3))
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_1113_order__antisym,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3))
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_1114_order__antisym,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_1115_order__antisym,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
       => ( X3 = Y ) ) ) ).

% order_antisym
tff(fact_1116_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_eq_set_int(A),C)) ) ) ).

% order.trans
tff(fact_1117_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% order.trans
tff(fact_1118_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% order.trans
tff(fact_1119_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% order.trans
tff(fact_1120_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% order.trans
tff(fact_1121_order__trans,axiom,
    ! [X3: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_eq_set_int(X3),Z)) ) ) ).

% order_trans
tff(fact_1122_order__trans,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Z)) ) ) ).

% order_trans
tff(fact_1123_order__trans,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Z)) ) ) ).

% order_trans
tff(fact_1124_order__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Z)) ) ) ).

% order_trans
tff(fact_1125_order__trans,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Z)) ) ) ).

% order_trans
tff(fact_1126_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A3),B2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B2)) )
     => ( ! [A3: rat,B2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B2),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B2)) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_1127_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A: num,B: num] :
      ( ! [A3: num,B2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A3),B2))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B2)) )
     => ( ! [A3: num,B2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(P,B2),A3))
           => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B2)) )
       => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_1128_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A3),B2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B2)) )
     => ( ! [A3: nat,B2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B2),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B2)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_1129_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),B2))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B2)) )
     => ( ! [A3: int,B2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P,B2),A3))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B2)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_1130_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_1131_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_1132_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_1133_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_1134_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_1135_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1136_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1137_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1138_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1139_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1140_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B))
       => pp(aa_set_int_bool(ord_less_eq_set_int(C),A)) ) ) ).

% dual_order.trans
tff(fact_1141_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% dual_order.trans
tff(fact_1142_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% dual_order.trans
tff(fact_1143_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% dual_order.trans
tff(fact_1144_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% dual_order.trans
tff(fact_1145_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_1146_antisym,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_1147_antisym,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_1148_antisym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_1149_antisym,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_1150_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_1151_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_1152_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_1153_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_1154_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_1155_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_1156_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_1157_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_1158_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => ( ! [X: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_1159_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_1160_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_1161_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_1162_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => ( ! [X: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X)),aa_int_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_1163_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_1164_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_1165_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1166_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1167_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1168_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1169_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1170_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1171_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1172_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1173_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1174_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,B)),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_1175_order__eq__refl,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( X3 = Y )
     => pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y)) ) ).

% order_eq_refl
tff(fact_1176_order__eq__refl,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 = Y )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y)) ) ).

% order_eq_refl
tff(fact_1177_order__eq__refl,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 = Y )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y)) ) ).

% order_eq_refl
tff(fact_1178_order__eq__refl,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ).

% order_eq_refl
tff(fact_1179_order__eq__refl,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 = Y )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y)) ) ).

% order_eq_refl
tff(fact_1180_linorder__linear,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3)) ) ).

% linorder_linear
tff(fact_1181_linorder__linear,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3)) ) ).

% linorder_linear
tff(fact_1182_linorder__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ).

% linorder_linear
tff(fact_1183_linorder__linear,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3)) ) ).

% linorder_linear
tff(fact_1184_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1185_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1186_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1187_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1188_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1189_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1190_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1191_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_num_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1192_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1193_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1194_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1195_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1196_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1197_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1198_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1199_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1200_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1201_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1202_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1203_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1204_linorder__le__cases,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3)) ) ).

% linorder_le_cases
tff(fact_1205_linorder__le__cases,axiom,
    ! [X3: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3)) ) ).

% linorder_le_cases
tff(fact_1206_linorder__le__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ).

% linorder_le_cases
tff(fact_1207_linorder__le__cases,axiom,
    ! [X3: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3)) ) ).

% linorder_le_cases
tff(fact_1208_order__antisym__conv,axiom,
    ! [Y: set_int,X3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_1209_order__antisym__conv,axiom,
    ! [Y: rat,X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_1210_order__antisym__conv,axiom,
    ! [Y: num,X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_1211_order__antisym__conv,axiom,
    ! [Y: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_1212_order__antisym__conv,axiom,
    ! [Y: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% order_antisym_conv
tff(fact_1213_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_1214_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_1215_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_1216_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_1217_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real)) ) ) ) ).

% mult_le_one
tff(fact_1218_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_1219_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ) ).

% mult_le_one
tff(fact_1220_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_1221_mult__right__le__one__le,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y)),X3)) ) ) ) ).

% mult_right_le_one_le
tff(fact_1222_mult__right__le__one__le,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y)),X3)) ) ) ) ).

% mult_right_le_one_le
tff(fact_1223_mult__right__le__one__le,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y)),X3)) ) ) ) ).

% mult_right_le_one_le
tff(fact_1224_mult__left__le__one__le,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X3)),X3)) ) ) ) ).

% mult_left_le_one_le
tff(fact_1225_mult__left__le__one__le,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X3)),X3)) ) ) ) ).

% mult_left_le_one_le
tff(fact_1226_mult__left__le__one__le,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X3)),X3)) ) ) ) ).

% mult_left_le_one_le
tff(fact_1227_lt__ex,axiom,
    ! [X3: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X3)) ).

% lt_ex
tff(fact_1228_lt__ex,axiom,
    ! [X3: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X3)) ).

% lt_ex
tff(fact_1229_lt__ex,axiom,
    ! [X3: int] :
    ? [Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X3)) ).

% lt_ex
tff(fact_1230_gt__ex,axiom,
    ! [X3: real] :
    ? [X_13: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),X_13)) ).

% gt_ex
tff(fact_1231_gt__ex,axiom,
    ! [X3: rat] :
    ? [X_13: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),X_13)) ).

% gt_ex
tff(fact_1232_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X_13)) ).

% gt_ex
tff(fact_1233_gt__ex,axiom,
    ! [X3: int] :
    ? [X_13: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),X_13)) ).

% gt_ex
tff(fact_1234_dense,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ? [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),Y)) ) ) ).

% dense
tff(fact_1235_dense,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ? [Z3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z3))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),Y)) ) ) ).

% dense
tff(fact_1236_less__imp__neq,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_1237_less__imp__neq,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_1238_less__imp__neq,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_1239_less__imp__neq,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_1240_less__imp__neq,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ( X3 != Y ) ) ).

% less_imp_neq
tff(fact_1241_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order.asym
tff(fact_1242_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order.asym
tff(fact_1243_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order.asym
tff(fact_1244_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).

% order.asym
tff(fact_1245_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% order.asym
tff(fact_1246_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_1247_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_1248_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_1249_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_1250_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_1251_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( B = C )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_1252_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_1253_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_1254_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_1255_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_1256_less__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [X: nat] :
          ( ! [Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),X))
             => pp(aa_nat_bool(P,Y4)) )
         => pp(aa_nat_bool(P,X)) )
     => pp(aa_nat_bool(P,A)) ) ).

% less_induct
tff(fact_1257_antisym__conv3,axiom,
    ! [Y: real,X3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_1258_antisym__conv3,axiom,
    ! [Y: rat,X3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_1259_antisym__conv3,axiom,
    ! [Y: num,X3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_1260_antisym__conv3,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_1261_antisym__conv3,axiom,
    ! [Y: int,X3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv3
tff(fact_1262_linorder__cases,axiom,
    ! [X3: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ( ( X3 != Y )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ) ).

% linorder_cases
tff(fact_1263_linorder__cases,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ( ( X3 != Y )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ) ).

% linorder_cases
tff(fact_1264_linorder__cases,axiom,
    ! [X3: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ( ( X3 != Y )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3)) ) ) ).

% linorder_cases
tff(fact_1265_linorder__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ( ( X3 != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ) ).

% linorder_cases
tff(fact_1266_linorder__cases,axiom,
    ! [X3: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ( ( X3 != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ) ).

% linorder_cases
tff(fact_1267_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% dual_order.asym
tff(fact_1268_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% dual_order.asym
tff(fact_1269_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% dual_order.asym
tff(fact_1270_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% dual_order.asym
tff(fact_1271_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% dual_order.asym
tff(fact_1272_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% dual_order.irrefl
tff(fact_1273_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% dual_order.irrefl
tff(fact_1274_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% dual_order.irrefl
tff(fact_1275_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% dual_order.irrefl
tff(fact_1276_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% dual_order.irrefl
tff(fact_1277_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
    <=> ? [N6: nat] :
          ( pp(aa_nat_bool(P,N6))
          & ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N6))
             => ~ pp(aa_nat_bool(P,M5)) ) ) ) ).

% exists_least_iff
tff(fact_1278_linorder__less__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,A: real,B: real] :
      ( ! [A3: real,B2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A3),B2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B2)) )
     => ( ! [A3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),A3))
       => ( ! [A3: real,B2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B2),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B2)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_1279_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A3),B2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B2)) )
     => ( ! [A3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),A3))
       => ( ! [A3: rat,B2: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B2),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B2)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_1280_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A: num,B: num] :
      ( ! [A3: num,B2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A3),B2))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B2)) )
     => ( ! [A3: num] : pp(aa_num_bool(aa_num_fun_num_bool(P,A3),A3))
       => ( ! [A3: num,B2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P,B2),A3))
             => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B2)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_1281_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B2)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),A3))
       => ( ! [A3: nat,B2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B2),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B2)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_1282_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B2))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B2)) )
     => ( ! [A3: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A3),A3))
       => ( ! [A3: int,B2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P,B2),A3))
             => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B2)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_1283_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans
tff(fact_1284_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans
tff(fact_1285_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans
tff(fact_1286_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans
tff(fact_1287_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans
tff(fact_1288_not__less__iff__gr__or__eq,axiom,
    ! [X3: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3))
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1289_not__less__iff__gr__or__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3))
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1290_not__less__iff__gr__or__eq,axiom,
    ! [X3: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3))
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1291_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3))
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1292_not__less__iff__gr__or__eq,axiom,
    ! [X3: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3))
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1293_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_1294_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_1295_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_1296_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_1297_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_1298_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1299_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1300_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1301_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1302_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1303_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1304_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1305_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1306_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1307_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1308_linorder__neqE,axiom,
    ! [X3: real,Y: real] :
      ( ( X3 != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ) ).

% linorder_neqE
tff(fact_1309_linorder__neqE,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ) ).

% linorder_neqE
tff(fact_1310_linorder__neqE,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 != Y )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3)) ) ) ).

% linorder_neqE
tff(fact_1311_linorder__neqE,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ) ).

% linorder_neqE
tff(fact_1312_linorder__neqE,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ) ).

% linorder_neqE
tff(fact_1313_order__less__asym,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ).

% order_less_asym
tff(fact_1314_order__less__asym,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ).

% order_less_asym
tff(fact_1315_order__less__asym,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3)) ) ).

% order_less_asym
tff(fact_1316_order__less__asym,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ).

% order_less_asym
tff(fact_1317_order__less__asym,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ).

% order_less_asym
tff(fact_1318_linorder__neq__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( X3 != Y )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ) ).

% linorder_neq_iff
tff(fact_1319_linorder__neq__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 != Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ) ).

% linorder_neq_iff
tff(fact_1320_linorder__neq__iff,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 != Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3)) ) ) ).

% linorder_neq_iff
tff(fact_1321_linorder__neq__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ) ).

% linorder_neq_iff
tff(fact_1322_linorder__neq__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 != Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ) ).

% linorder_neq_iff
tff(fact_1323_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order_less_asym'
tff(fact_1324_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order_less_asym'
tff(fact_1325_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order_less_asym'
tff(fact_1326_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).

% order_less_asym'
tff(fact_1327_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% order_less_asym'
tff(fact_1328_order__less__trans,axiom,
    ! [X3: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z)) ) ) ).

% order_less_trans
tff(fact_1329_order__less__trans,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z)) ) ) ).

% order_less_trans
tff(fact_1330_order__less__trans,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z)) ) ) ).

% order_less_trans
tff(fact_1331_order__less__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z)) ) ) ).

% order_less_trans
tff(fact_1332_order__less__trans,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z)) ) ) ).

% order_less_trans
tff(fact_1333_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1334_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1335_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_real_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1336_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1337_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1338_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1339_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1340_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1341_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1342_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_1343_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1344_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1345_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1346_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1347_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1348_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1349_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1350_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1351_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1352_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_1353_order__less__irrefl,axiom,
    ! [X3: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),X3)) ).

% order_less_irrefl
tff(fact_1354_order__less__irrefl,axiom,
    ! [X3: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),X3)) ).

% order_less_irrefl
tff(fact_1355_order__less__irrefl,axiom,
    ! [X3: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),X3)) ).

% order_less_irrefl
tff(fact_1356_order__less__irrefl,axiom,
    ! [X3: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X3)) ).

% order_less_irrefl
tff(fact_1357_order__less__irrefl,axiom,
    ! [X3: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),X3)) ).

% order_less_irrefl
tff(fact_1358_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1359_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1360_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1361_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1362_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1363_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1364_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1365_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1366_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1367_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_1368_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1369_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,B)),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1370_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,B)),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1371_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B)),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1372_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,B)),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1373_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1374_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1375_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1376_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1377_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_1378_order__less__not__sym,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ).

% order_less_not_sym
tff(fact_1379_order__less__not__sym,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ).

% order_less_not_sym
tff(fact_1380_order__less__not__sym,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3)) ) ).

% order_less_not_sym
tff(fact_1381_order__less__not__sym,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ).

% order_less_not_sym
tff(fact_1382_order__less__not__sym,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ).

% order_less_not_sym
tff(fact_1383_order__less__imp__triv,axiom,
    ! [X3: real,Y: real,P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_1384_order__less__imp__triv,axiom,
    ! [X3: rat,Y: rat,P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_1385_order__less__imp__triv,axiom,
    ! [X3: num,Y: num,P: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_1386_order__less__imp__triv,axiom,
    ! [X3: nat,Y: nat,P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_1387_order__less__imp__triv,axiom,
    ! [X3: int,Y: int,P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_1388_linorder__less__linear,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
      | ( X3 = Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ).

% linorder_less_linear
tff(fact_1389_linorder__less__linear,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
      | ( X3 = Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ).

% linorder_less_linear
tff(fact_1390_linorder__less__linear,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
      | ( X3 = Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3)) ) ).

% linorder_less_linear
tff(fact_1391_linorder__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
      | ( X3 = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ).

% linorder_less_linear
tff(fact_1392_linorder__less__linear,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
      | ( X3 = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ).

% linorder_less_linear
tff(fact_1393_order__less__imp__not__eq,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1394_order__less__imp__not__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1395_order__less__imp__not__eq,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1396_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1397_order__less__imp__not__eq,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1398_order__less__imp__not__eq2,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1399_order__less__imp__not__eq2,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1400_order__less__imp__not__eq2,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1401_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1402_order__less__imp__not__eq2,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
tff(fact_1403_order__less__imp__not__less,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ).

% order_less_imp_not_less
tff(fact_1404_order__less__imp__not__less,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ).

% order_less_imp_not_less
tff(fact_1405_order__less__imp__not__less,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3)) ) ).

% order_less_imp_not_less
tff(fact_1406_order__less__imp__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ).

% order_less_imp_not_less
tff(fact_1407_order__less__imp__not__less,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ).

% order_less_imp_not_less
tff(fact_1408_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3),S2) )
     => ord_le3146513528884898305at_nat(R2,S2) ) ).

% subrelI
tff(fact_1409_subrelI,axiom,
    ! [R2: set_Pr7556676689462069481BT_nat,S2: set_Pr7556676689462069481BT_nat] :
      ( ! [X: vEBT_VEBT,Y3: nat] :
          ( member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X),Y3),R2)
         => member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X),Y3),S2) )
     => ord_le3442269383143156041BT_nat(R2,S2) ) ).

% subrelI
tff(fact_1410_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X: int,Y3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y3),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y3),S2) )
     => ord_le2843351958646193337nt_int(R2,S2) ) ).

% subrelI
tff(fact_1411_subrelI,axiom,
    ! [R2: set_Pr4811707699266497531nteger,S2: set_Pr4811707699266497531nteger] :
      ( ! [X: code_integer,Y3: code_integer] :
          ( member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X),Y3),R2)
         => member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X),Y3),S2) )
     => ord_le3725938330318615451nteger(R2,S2) ) ).

% subrelI
tff(fact_1412_subrelI,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
      ( ! [X: nat,Y3: num] :
          ( member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X),Y3),R2)
         => member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X),Y3),S2) )
     => ord_le8085105155179020875at_num(R2,S2) ) ).

% subrelI
tff(fact_1413_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_1414_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X3: nat,M8: nat] :
      ( pp(aa_nat_bool(P,X3))
     => ( ! [X: nat] :
            ( pp(aa_nat_bool(P,X))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),M8)) )
       => ~ ! [M2: nat] :
              ( pp(aa_nat_bool(P,M2))
             => ~ ! [X5: nat] :
                    ( pp(aa_nat_bool(P,X5))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),M2)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1415_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),X3))
    <=> ( ( ( X3 = zero_zero_nat )
         => pp(A) )
        & ( ( X3 != zero_zero_nat )
         => ( ( ( X3 = one_one_nat )
             => pp(B) )
            & ( X3 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_1416_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),X3))
    <=> ( ( ( X3 = zero_zero_nat )
         => pp(A) )
        & ( ( X3 != zero_zero_nat )
         => ( ( ( X3 = one_one_nat )
             => pp(B) )
            & ( X3 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_1417_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1418_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1419_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1420_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1421_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1422_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1423_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1424_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1425_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1426_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1427_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1428_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1429_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1430_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1431_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1432_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1433_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1434_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1435_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1436_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1437_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1438_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1439_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1440_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1441_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le4899916226251588169t_bool(aa_set1256819327649670340t_bool(aTP_Lamm_dp,R),aa_set1256819327649670340t_bool(aTP_Lamm_dp,S3))
    <=> ord_le3146513528884898305at_nat(R,S3) ) ).

% pred_subset_eq2
tff(fact_1442_pred__subset__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat] :
      ( ord_le3072161858217682703t_bool(aa_set2648297115105225046t_bool(aTP_Lamm_dq,R),aa_set2648297115105225046t_bool(aTP_Lamm_dq,S3))
    <=> ord_le3442269383143156041BT_nat(R,S3) ) ).

% pred_subset_eq2
tff(fact_1443_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le4546832483993983377t_bool(aa_set8965632145344517972t_bool(aTP_Lamm_dr,R),aa_set8965632145344517972t_bool(aTP_Lamm_dr,S3))
    <=> ord_le2843351958646193337nt_int(R,S3) ) ).

% pred_subset_eq2
tff(fact_1444_pred__subset__eq2,axiom,
    ! [R: set_Pr4811707699266497531nteger,S3: set_Pr4811707699266497531nteger] :
      ( ord_le2895103514575585327r_bool(aa_set8960070006189174672r_bool(aTP_Lamm_ds,R),aa_set8960070006189174672r_bool(aTP_Lamm_ds,S3))
    <=> ord_le3725938330318615451nteger(R,S3) ) ).

% pred_subset_eq2
tff(fact_1445_pred__subset__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ord_le8094096262961209215m_bool(aa_set4109929515018738736m_bool(aTP_Lamm_dt,R),aa_set4109929515018738736m_bool(aTP_Lamm_dt,S3))
    <=> ord_le8085105155179020875at_num(R,S3) ) ).

% pred_subset_eq2
tff(fact_1446_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Xa3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3),R)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_1447_pred__equals__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat] :
      ( ! [X4: vEBT_VEBT,Xa3: nat] :
          ( member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X4),Xa3),R)
        <=> member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X4),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_1448_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Xa3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X4),Xa3),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X4),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_1449_pred__equals__eq2,axiom,
    ! [R: set_Pr4811707699266497531nteger,S3: set_Pr4811707699266497531nteger] :
      ( ! [X4: code_integer,Xa3: code_integer] :
          ( member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X4),Xa3),R)
        <=> member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X4),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_1450_pred__equals__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ! [X4: nat,Xa3: num] :
          ( member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X4),Xa3),R)
        <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X4),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_1451_vebt__mint_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] :
      ( ( pp(A)
       => ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
      & ( ~ pp(A)
       => ( ( pp(B)
           => ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = aa_nat_option_nat(some_nat,one_one_nat) ) )
          & ( ~ pp(B)
           => ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
tff(fact_1452_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: bool,A: bool] :
      ( ( pp(B)
       => ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = aa_nat_option_nat(some_nat,one_one_nat) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
          & ( ~ pp(A)
           => ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
tff(fact_1453_vebt__succ_Osimps_I1_J,axiom,
    ! [B: bool,Uu: bool] :
      ( ( pp(B)
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),B)),zero_zero_nat) = aa_nat_option_nat(some_nat,one_one_nat) ) )
      & ( ~ pp(B)
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),B)),zero_zero_nat) = none_nat ) ) ) ).

% vebt_succ.simps(1)
tff(fact_1454_leD,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ).

% leD
tff(fact_1455_leD,axiom,
    ! [Y: set_int,X3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X3))
     => ~ pp(aa_set_int_bool(ord_less_set_int(X3),Y)) ) ).

% leD
tff(fact_1456_leD,axiom,
    ! [Y: rat,X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y)) ) ).

% leD
tff(fact_1457_leD,axiom,
    ! [Y: num,X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y)) ) ).

% leD
tff(fact_1458_leD,axiom,
    ! [Y: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y)) ) ).

% leD
tff(fact_1459_leD,axiom,
    ! [Y: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y)) ) ).

% leD
tff(fact_1460_leI,axiom,
    ! [X3: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3)) ) ).

% leI
tff(fact_1461_leI,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3)) ) ).

% leI
tff(fact_1462_leI,axiom,
    ! [X3: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3)) ) ).

% leI
tff(fact_1463_leI,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ).

% leI
tff(fact_1464_leI,axiom,
    ! [X3: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3)) ) ).

% leI
tff(fact_1465_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1466_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(A),B))
    <=> ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1467_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1468_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1469_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1470_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1471_antisym__conv1,axiom,
    ! [X3: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1472_antisym__conv1,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ~ pp(aa_set_int_bool(ord_less_set_int(X3),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1473_antisym__conv1,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1474_antisym__conv1,axiom,
    ! [X3: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1475_antisym__conv1,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1476_antisym__conv1,axiom,
    ! [X3: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv1
tff(fact_1477_antisym__conv2,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1478_antisym__conv2,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
     => ( ~ pp(aa_set_int_bool(ord_less_set_int(X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1479_antisym__conv2,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1480_antisym__conv2,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1481_antisym__conv2,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1482_antisym__conv2,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
      <=> ( X3 = Y ) ) ) ).

% antisym_conv2
tff(fact_1483_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_ge
tff(fact_1484_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_ge
tff(fact_1485_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Z)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ).

% dense_le
tff(fact_1486_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ).

% dense_le
tff(fact_1487_less__le__not__le,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3)) ) ) ).

% less_le_not_le
tff(fact_1488_less__le__not__le,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X3),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(Y),X3)) ) ) ).

% less_le_not_le
tff(fact_1489_less__le__not__le,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3)) ) ) ).

% less_le_not_le
tff(fact_1490_less__le__not__le,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3)) ) ) ).

% less_le_not_le
tff(fact_1491_less__le__not__le,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ) ).

% less_le_not_le
tff(fact_1492_less__le__not__le,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3)) ) ) ).

% less_le_not_le
tff(fact_1493_not__le__imp__less,axiom,
    ! [Y: real,X3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ).

% not_le_imp_less
tff(fact_1494_not__le__imp__less,axiom,
    ! [Y: rat,X3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y)) ) ).

% not_le_imp_less
tff(fact_1495_not__le__imp__less,axiom,
    ! [Y: num,X3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y)) ) ).

% not_le_imp_less
tff(fact_1496_not__le__imp__less,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y)) ) ).

% not_le_imp_less
tff(fact_1497_not__le__imp__less,axiom,
    ! [Y: int,X3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y)) ) ).

% not_le_imp_less
tff(fact_1498_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1499_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1500_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1501_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1502_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1503_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1504_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1505_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1506_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1507_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1508_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1509_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1510_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans1
tff(fact_1511_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_set_int(A),C)) ) ) ).

% order.strict_trans1
tff(fact_1512_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_1513_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans1
tff(fact_1514_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_1515_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans1
tff(fact_1516_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans2
tff(fact_1517_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),C))
       => pp(aa_set_int_bool(ord_less_set_int(A),C)) ) ) ).

% order.strict_trans2
tff(fact_1518_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_1519_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans2
tff(fact_1520_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_1521_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans2
tff(fact_1522_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_1523_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ) ).

% order.strict_iff_not
tff(fact_1524_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_1525_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_1526_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_1527_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_1528_dense__ge__bounded,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X3))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),X3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),W)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_1529_dense__ge__bounded,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X3))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),X3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),W)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_ge_bounded
tff(fact_1530_dense__le__bounded,axiom,
    ! [X3: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Y))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_1531_dense__le__bounded,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Y))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% dense_le_bounded
tff(fact_1532_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1533_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1534_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1535_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1536_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1537_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1538_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1539_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1540_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1541_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1542_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1543_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1544_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_1545_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_set_int(C),B))
       => pp(aa_set_int_bool(ord_less_set_int(C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_1546_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_1547_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_1548_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_1549_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_1550_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_1551_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B))
       => pp(aa_set_int_bool(ord_less_set_int(C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_1552_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_1553_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_1554_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_1555_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_1556_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1557_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1558_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1559_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1560_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1561_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1562_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% order.strict_implies_order
tff(fact_1563_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A),B)) ) ).

% order.strict_implies_order
tff(fact_1564_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% order.strict_implies_order
tff(fact_1565_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% order.strict_implies_order
tff(fact_1566_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% order.strict_implies_order
tff(fact_1567_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% order.strict_implies_order
tff(fact_1568_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_1569_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(B),A))
     => pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_1570_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_1571_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_1572_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_1573_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_1574_order__le__less,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1575_order__le__less,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
    <=> ( pp(aa_set_int_bool(ord_less_set_int(X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1576_order__le__less,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1577_order__le__less,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1578_order__le__less,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1579_order__le__less,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_less
tff(fact_1580_order__less__le,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1581_order__less__le,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X3),Y))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1582_order__less__le,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1583_order__less__le,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1584_order__less__le,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1585_order__less__le,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
        & ( X3 != Y ) ) ) ).

% order_less_le
tff(fact_1586_linorder__not__le,axiom,
    ! [X3: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ).

% linorder_not_le
tff(fact_1587_linorder__not__le,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ).

% linorder_not_le
tff(fact_1588_linorder__not__le,axiom,
    ! [X3: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3)) ) ).

% linorder_not_le
tff(fact_1589_linorder__not__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ).

% linorder_not_le
tff(fact_1590_linorder__not__le,axiom,
    ! [X3: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ).

% linorder_not_le
tff(fact_1591_linorder__not__less,axiom,
    ! [X3: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3)) ) ).

% linorder_not_less
tff(fact_1592_linorder__not__less,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3)) ) ).

% linorder_not_less
tff(fact_1593_linorder__not__less,axiom,
    ! [X3: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3)) ) ).

% linorder_not_less
tff(fact_1594_linorder__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ).

% linorder_not_less
tff(fact_1595_linorder__not__less,axiom,
    ! [X3: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3)) ) ).

% linorder_not_less
tff(fact_1596_order__less__imp__le,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ).

% order_less_imp_le
tff(fact_1597_order__less__imp__le,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X3),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y)) ) ).

% order_less_imp_le
tff(fact_1598_order__less__imp__le,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y)) ) ).

% order_less_imp_le
tff(fact_1599_order__less__imp__le,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y)) ) ).

% order_less_imp_le
tff(fact_1600_order__less__imp__le,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ).

% order_less_imp_le
tff(fact_1601_order__less__imp__le,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y)) ) ).

% order_less_imp_le
tff(fact_1602_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ( A != B )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_1603_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( ( A != B )
       => pp(aa_set_int_bool(ord_less_set_int(A),B)) ) ) ).

% order_le_neq_trans
tff(fact_1604_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( A != B )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_1605_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( A != B )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_1606_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( A != B )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_1607_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( A != B )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_1608_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_1609_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => pp(aa_set_int_bool(ord_less_set_int(A),B)) ) ) ).

% order_neq_le_trans
tff(fact_1610_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_1611_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_1612_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_1613_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_1614_order__le__less__trans,axiom,
    ! [X3: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z)) ) ) ).

% order_le_less_trans
tff(fact_1615_order__le__less__trans,axiom,
    ! [X3: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X3),Z)) ) ) ).

% order_le_less_trans
tff(fact_1616_order__le__less__trans,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z)) ) ) ).

% order_le_less_trans
tff(fact_1617_order__le__less__trans,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z)) ) ) ).

% order_le_less_trans
tff(fact_1618_order__le__less__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z)) ) ) ).

% order_le_less_trans
tff(fact_1619_order__le__less__trans,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z)) ) ) ).

% order_le_less_trans
tff(fact_1620_order__less__le__trans,axiom,
    ! [X3: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z)) ) ) ).

% order_less_le_trans
tff(fact_1621_order__less__le__trans,axiom,
    ! [X3: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(X3),Y))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z))
       => pp(aa_set_int_bool(ord_less_set_int(X3),Z)) ) ) ).

% order_less_le_trans
tff(fact_1622_order__less__le__trans,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z)) ) ) ).

% order_less_le_trans
tff(fact_1623_order__less__le__trans,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z)) ) ) ).

% order_less_le_trans
tff(fact_1624_order__less__le__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z)) ) ) ).

% order_less_le_trans
tff(fact_1625_order__less__le__trans,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z)) ) ) ).

% order_less_le_trans
tff(fact_1626_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1627_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1628_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1629_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1630_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1631_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1632_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1633_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1634_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1635_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1636_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1637_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1638_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1639_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1640_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1641_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1642_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1643_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1644_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1645_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1646_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1647_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1648_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1649_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1650_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1651_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1652_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1653_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1654_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1655_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1656_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1657_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1658_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1659_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B)),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1660_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B)),C))
       => ( ! [X: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1661_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,B)),C))
       => ( ! [X: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1662_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C))
       => ( ! [X: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1663_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
       => ( ! [X: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1664_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
       => ( ! [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1665_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,B)),C))
       => ( ! [X: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1666_linorder__le__less__linear,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ).

% linorder_le_less_linear
tff(fact_1667_linorder__le__less__linear,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X3)) ) ).

% linorder_le_less_linear
tff(fact_1668_linorder__le__less__linear,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X3)) ) ).

% linorder_le_less_linear
tff(fact_1669_linorder__le__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X3)) ) ).

% linorder_le_less_linear
tff(fact_1670_linorder__le__less__linear,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X3)) ) ).

% linorder_le_less_linear
tff(fact_1671_order__le__imp__less__or__eq,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1672_order__le__imp__less__or__eq,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
     => ( pp(aa_set_int_bool(ord_less_set_int(X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1673_order__le__imp__less__or__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1674_order__le__imp__less__or__eq,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1675_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1676_order__le__imp__less__or__eq,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Y))
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1677_bot_Oextremum,axiom,
    ! [A: set_real] : pp(aa_set_real_bool(ord_less_eq_set_real(bot_bot_set_real),A)) ).

% bot.extremum
tff(fact_1678_bot_Oextremum,axiom,
    ! [A: set_bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(bot_bot_set_bool),A)) ).

% bot.extremum
tff(fact_1679_bot_Oextremum,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,bot_bot_set_nat),A)) ).

% bot.extremum
tff(fact_1680_bot_Oextremum,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A)) ).

% bot.extremum
tff(fact_1681_bot_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,bot_bot_nat),A)) ).

% bot.extremum
tff(fact_1682_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_1683_bot_Oextremum__unique,axiom,
    ! [A: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A),bot_bot_set_bool))
    <=> ( A = bot_bot_set_bool ) ) ).

% bot.extremum_unique
tff(fact_1684_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),bot_bot_set_nat))
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_1685_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_1686_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_1687_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A),bot_bot_set_real))
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_1688_bot_Oextremum__uniqueI,axiom,
    ! [A: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A),bot_bot_set_bool))
     => ( A = bot_bot_set_bool ) ) ).

% bot.extremum_uniqueI
tff(fact_1689_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),bot_bot_set_nat))
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1690_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_1691_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1692_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A),bot_bot_set_real)) ).

% bot.extremum_strict
tff(fact_1693_bot_Oextremum__strict,axiom,
    ! [A: set_bool] : ~ pp(aa_set_bool_bool(ord_less_set_bool(A),bot_bot_set_bool)) ).

% bot.extremum_strict
tff(fact_1694_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),bot_bot_set_nat)) ).

% bot.extremum_strict
tff(fact_1695_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A),bot_bot_set_int)) ).

% bot.extremum_strict
tff(fact_1696_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_1697_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_set_real(bot_bot_set_real),A)) ) ).

% bot.not_eq_extremum
tff(fact_1698_bot_Onot__eq__extremum,axiom,
    ! [A: set_bool] :
      ( ( A != bot_bot_set_bool )
    <=> pp(aa_set_bool_bool(ord_less_set_bool(bot_bot_set_bool),A)) ) ).

% bot.not_eq_extremum
tff(fact_1699_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,bot_bot_set_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1700_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_set_int(bot_bot_set_int),A)) ) ).

% bot.not_eq_extremum
tff(fact_1701_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1702_vebt__pred_Osimps_I3_J,axiom,
    ! [B: bool,A: bool,Va2: nat] :
      ( ( pp(B)
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_option_nat(some_nat,one_one_nat) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( aa_nat_option_nat(vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
          & ( ~ pp(A)
           => ( aa_nat_option_nat(vEBT_vebt_pred(aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_1703_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N: nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),N7))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N7)) ) ).

% bounded_nat_set_is_finite
tff(fact_1704_finite__nat__set__iff__bounded,axiom,
    ! [N7: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N7))
    <=> ? [M5: nat] :
        ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),N7))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),M5)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1705_finite__nat__set__iff__bounded__le,axiom,
    ! [N7: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N7))
    <=> ? [M5: nat] :
        ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),N7))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M5)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1706_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : pp(aa_nat_bool(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_maxti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_av(T),Ti)),one_one_nat)) ).

% vebt_maxti_hT
tff(fact_1707_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : pp(aa_nat_bool(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_minti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_be(T),Ti)),one_one_nat)) ).

% vebt_minti_hT
tff(fact_1708_field__le__mult__one__interval,axiom,
    ! [X3: real,Y: real] :
      ( ! [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z3),X3)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ).

% field_le_mult_one_interval
tff(fact_1709_field__le__mult__one__interval,axiom,
    ! [X3: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z3))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),X3)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y)) ) ).

% field_le_mult_one_interval
tff(fact_1710_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X3: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_du(I5),X3))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_du(I5),Y))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_dv(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1711_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dw(I5),X3))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dw(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_dx(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1712_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dy(I5),X3))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dy(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_dz(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1713_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ea(I5),X3))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ea(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_eb(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1714_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ec(I5),X3))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ec(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_ed(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1715_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X3: fun_Co3960896333048504454r_real,Y: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun7028086442024493327r_bool(aTP_Lamm_ee(I5),X3))))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun7028086442024493327r_bool(aTP_Lamm_ee(I5),Y))))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun7028086442024493327r_bool(aa_fun7037952363265197026r_bool(aTP_Lamm_ef(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1716_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X3: fun_VEBT_VEBT_rat,Y: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1017860439744518025T_bool(aTP_Lamm_eg(I5),X3))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1017860439744518025T_bool(aTP_Lamm_eg(I5),Y))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1017860439744518025T_bool(aa_fun5042198764111188021T_bool(aTP_Lamm_eh(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1717_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: fun_real_rat,Y: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_ei(I5),X3))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_ei(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_ej(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1718_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: fun_nat_rat,Y: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ek(I5),X3))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ek(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_el(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1719_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: fun_int_rat,Y: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_em(I5),X3))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_em(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aa_fun465274253410646749t_bool(aTP_Lamm_en(I5),X3),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_1720_less__numeral__extra_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% less_numeral_extra(1)
tff(fact_1721_less__numeral__extra_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% less_numeral_extra(1)
tff(fact_1722_less__numeral__extra_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% less_numeral_extra(1)
tff(fact_1723_less__numeral__extra_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% less_numeral_extra(1)
tff(fact_1724_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),X3) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_1725_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),X3) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_1726_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),X3) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_1727_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,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,X3) = Y )
     => ( ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2)
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue)
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_1728_option_Osize__gen_I1_J,axiom,
    ! [X3: fun_nat_nat] : size_option_nat(X3,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1729_option_Osize__gen_I1_J,axiom,
    ! [X3: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X3,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1730_option_Osize__gen_I1_J,axiom,
    ! [X3: fun_num_nat] : size_option_num(X3,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1731_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_1732_pure__true,axiom,
    aa_bool_assn(pure_assn,fTrue) = one_one_assn ).

% pure_true
tff(fact_1733_pure__assn__eq__emp__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_assn(pure_assn,P) = one_one_assn )
    <=> pp(P) ) ).

% pure_assn_eq_emp_iff
tff(fact_1734_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_1735_norm__pre__pure__iff__sng,axiom,
    ! [B: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(aa_bool_assn(pure_assn,B),F,Q)
    <=> ( pp(B)
       => hoare_7629718768684598413on_nat(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_1736_norm__pre__pure__iff__sng,axiom,
    ! [B: bool,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( hoare_2774773592644289642e_bool(aa_bool_assn(pure_assn,B),F,Q)
    <=> ( pp(B)
       => hoare_2774773592644289642e_bool(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_1737_norm__pre__pure__iff__sng,axiom,
    ! [B: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(aa_bool_assn(pure_assn,B),F,Q)
    <=> ( pp(B)
       => hoare_1429296392585015714_VEBTi(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_1738_norm__pre__pure__iff__sng,axiom,
    ! [B: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(aa_bool_assn(pure_assn,B),F,Q)
    <=> ( pp(B)
       => hoare_3067605981109127869le_nat(one_one_assn,F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_1739_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% norm_assertion_simps(2)
tff(fact_1740_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% norm_assertion_simps(1)
tff(fact_1741_assn__one__left,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),P) = P ).

% assn_one_left
tff(fact_1742_norm__pre__pure__rule2,axiom,
    ! [B: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ( pp(B)
       => hoare_7629718768684598413on_nat(one_one_assn,F,Q) )
     => hoare_7629718768684598413on_nat(aa_bool_assn(pure_assn,B),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_1743_norm__pre__pure__rule2,axiom,
    ! [B: bool,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( ( pp(B)
       => hoare_2774773592644289642e_bool(one_one_assn,F,Q) )
     => hoare_2774773592644289642e_bool(aa_bool_assn(pure_assn,B),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_1744_norm__pre__pure__rule2,axiom,
    ! [B: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ( pp(B)
       => hoare_1429296392585015714_VEBTi(one_one_assn,F,Q) )
     => hoare_1429296392585015714_VEBTi(aa_bool_assn(pure_assn,B),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_1745_norm__pre__pure__rule2,axiom,
    ! [B: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ( pp(B)
       => hoare_3067605981109127869le_nat(one_one_assn,F,Q) )
     => hoare_3067605981109127869le_nat(aa_bool_assn(pure_assn,B),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_1746_linordered__field__no__ub,axiom,
    ! [X5: real] :
    ? [X_13: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),X_13)) ).

% linordered_field_no_ub
tff(fact_1747_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_13: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),X_13)) ).

% linordered_field_no_ub
tff(fact_1748_linordered__field__no__lb,axiom,
    ! [X5: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X5)) ).

% linordered_field_no_lb
tff(fact_1749_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X5)) ).

% linordered_field_no_lb
tff(fact_1750_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1751_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),fTrue)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_1752_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_1753_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,aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_1754_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: bool,B: bool,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),X3) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_1755_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: bool,B: bool,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),aa_nat_nat(suc,aa_nat_nat(suc,N))) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_1756_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: bool,B: bool,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),aa_nat_nat(suc,aa_nat_nat(suc,N))) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_1757_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: bool,B: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_bool_VEBT_VEBT(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_1758_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: bool,B: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_bool_VEBT_VEBT(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_1759_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_1760_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_1761_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_bool_VEBT_VEBT(vEBT_Leaf(Uv),Uw)),aa_nat_nat(suc,N)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_1762_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: bool,B: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),B)),zero_zero_nat) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_1763_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_1764_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(aa_Pro4773402355715522498at_nat(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_1765_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_1766_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),X3) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_1767_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: bool,B: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_bool_VEBT_VEBT(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_1768_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: bool,B: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_bool_VEBT_VEBT(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_1769_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),zero_zero_nat,Vc),Vd)),Ve) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_1770_le__numeral__extra_I3_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_zero_real)) ).

% le_numeral_extra(3)
tff(fact_1771_le__numeral__extra_I3_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_zero_rat)) ).

% le_numeral_extra(3)
tff(fact_1772_le__numeral__extra_I3_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).

% le_numeral_extra(3)
tff(fact_1773_le__numeral__extra_I3_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% le_numeral_extra(3)
tff(fact_1774_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_zero_real)) ).

% less_numeral_extra(3)
tff(fact_1775_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_zero_rat)) ).

% less_numeral_extra(3)
tff(fact_1776_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).

% less_numeral_extra(3)
tff(fact_1777_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_numeral_extra(3)
tff(fact_1778_le__numeral__extra_I4_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),one_one_real)) ).

% le_numeral_extra(4)
tff(fact_1779_le__numeral__extra_I4_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),one_one_rat)) ).

% le_numeral_extra(4)
tff(fact_1780_le__numeral__extra_I4_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).

% le_numeral_extra(4)
tff(fact_1781_le__numeral__extra_I4_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).

% le_numeral_extra(4)
tff(fact_1782_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),one_one_real)) ).

% less_numeral_extra(4)
tff(fact_1783_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),one_one_rat)) ).

% less_numeral_extra(4)
tff(fact_1784_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).

% less_numeral_extra(4)
tff(fact_1785_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).

% less_numeral_extra(4)
tff(fact_1786_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),zero_zero_nat,Uy),Uz)),X3) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_1787_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList),Summary)),X3) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_1788_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList),Summary)),X3) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_1789_builupi_Hcorr,axiom,
    ! [N: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))) ).

% builupi'corr
tff(fact_1790_builupicorr,axiom,
    ! [N: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))) ).

% builupicorr
tff(fact_1791_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_option_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_1792_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_1793_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: bool,F: heap_Time_Heap_bool,Q: fun_bool_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_bool(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_1794_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_1795_norm__pre__pure__iff__htt_H,axiom,
    ! [B: bool,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,B)),P),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_option_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_1796_norm__pre__pure__iff__htt_H,axiom,
    ! [B: bool,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,B)),P),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_1797_norm__pre__pure__iff__htt_H,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_bool,Q: fun_bool_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,B)),P),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_bool(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_1798_norm__pre__pure__iff__htt_H,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,B)),P),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_1799_Collect__empty__eq__bot,axiom,
    ! [P: fun_complex_bool] :
      ( ( collect_complex(P) = bot_bot_set_complex )
    <=> ( P = bot_bo8490223647127498926x_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1800_Collect__empty__eq__bot,axiom,
    ! [P: fun_list_nat_bool] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ( P = bot_bo6469247547558333820t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1801_Collect__empty__eq__bot,axiom,
    ! [P: fun_set_nat_bool] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ( P = bot_bo8343409700046576086t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1802_Collect__empty__eq__bot,axiom,
    ! [P: fun_real_bool] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ( P = bot_bo4181092155253064880l_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1803_Collect__empty__eq__bot,axiom,
    ! [P: fun_bool_bool] :
      ( ( collect_bool(P) = bot_bot_set_bool )
    <=> ( P = bot_bo4756488196683355415l_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1804_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1805_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_1806_htt__htD,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(P,C,Q),T))
     => hoare_7629718768684598413on_nat(P,C,Q) ) ).

% htt_htD
tff(fact_1807_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_bool(P,C,Q),T))
     => hoare_2774773592644289642e_bool(P,C,Q) ) ).

% htt_htD
tff(fact_1808_htt__htD,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,C,Q),T))
     => hoare_1429296392585015714_VEBTi(P,C,Q) ) ).

% htt_htD
tff(fact_1809_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(P,C,Q),T))
     => hoare_3067605981109127869le_nat(P,C,Q) ) ).

% htt_htD
tff(fact_1810_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_1811_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_1812_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_1813_htt__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% htt_vebt_buildupi
tff(fact_1814_htt__vebt__buildupi_H,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% htt_vebt_buildupi'
tff(fact_1815_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,Va2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_bool_VEBT_VEBT(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_1816_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),Uv)),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_1817_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),Uv)),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_1818_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_bool_VEBT_VEBT(vEBT_Leaf(Uv),Uw)),aa_nat_nat(suc,N)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_1819_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va2)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_1820_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va2)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_1821_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_1822_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: bool,Uw: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_bool_VEBT_VEBT(vEBT_Leaf(A),Uw)),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_1823_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),zero_zero_nat,Vd),Ve)),Vf) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_1824_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),zero_zero_nat,Vd),Ve)),Vf) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_1825_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),zero_zero_nat,Vc),Vd)),Ve) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_1826_nat__descend__induct,axiom,
    ! [N: nat,P: fun_nat_bool,M: nat] :
      ( ! [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K2))
         => pp(aa_nat_bool(P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
           => ( ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),I4))
                 => pp(aa_nat_bool(P,I4)) )
             => pp(aa_nat_bool(P,K2)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_1827_T__vebt__buildupi,axiom,
    ! [N: nat,H: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(vEBT_V739175172307565963ildupi(N),H)),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% T_vebt_buildupi
tff(fact_1828_vebt__buildupi__refines,axiom,
    ! [N: nat] : refine5565527176597971370_VEBTi(vEBT_vebt_buildupi(N),vEBT_V739175172307565963ildupi(N)) ).

% vebt_buildupi_refines
tff(fact_1829_TBOUND__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_V739175172307565963ildupi(N)),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% TBOUND_vebt_buildupi
tff(fact_1830_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ! [M5: nat] :
        ? [N6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N6))
          & pp(aa_set_nat_bool(member_nat(N6),S3)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1831_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M2))
         => ? [N8: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N8))
              & pp(aa_set_nat_bool(member_nat(N8),S3)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S3)) ) ).

% unbounded_k_infinite
tff(fact_1832_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ! [M5: nat] :
        ? [N6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N6))
          & pp(aa_set_nat_bool(member_nat(N6),S3)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1833_VEBT__internal_Oheight_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ! [A3: bool,B2: bool] : X3 != aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2) ) ).

% VEBT_internal.height.cases
tff(fact_1834_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_vebt_minti(T)),one_one_nat)) ).

% TBOUND_vebt_minti
tff(fact_1835_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_vebt_maxti(T)),one_one_nat)) ).

% TBOUND_vebt_maxti
tff(fact_1836_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X5)),aa_complex_real(F,lattic8794016678065449205x_real(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1837_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ~ ? [X5: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X5),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Code_integer_real(F,X5)),aa_Code_integer_real(F,lattic2659822949269061924r_real(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1838_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X5)),aa_real_real(F,lattic8440615504127631091l_real(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1839_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_bool,F: fun_bool_real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ( ( S3 != bot_bot_set_bool )
       => ~ ? [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_bool_real(F,X5)),aa_bool_real(F,aa_set_bool_bool(lattic5237676299781464426l_real(F),S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1840_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X5)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1841_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X5)),aa_int_real(F,aa_set_int_int(lattic2675449441010098035t_real(F),S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1842_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X5: complex] :
              ( pp(aa_set_complex_bool(member_complex(X5),S3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X5)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1843_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ~ ? [X5: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X5),S3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Code_integer_rat(F,X5)),aa_Code_integer_rat(F,lattic5439806495466278992er_rat(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1844_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),S3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X5)),aa_real_rat(F,lattic4420706379359479199al_rat(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1845_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ( ( S3 != bot_bot_set_bool )
       => ~ ? [X5: bool] :
              ( pp(aa_set_bool_bool(member_bool(X5),S3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_bool_rat(F,X5)),aa_bool_rat(F,aa_set_bool_bool(lattic6202137068483785622ol_rat(F),S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1846_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: fun_bo851793448214054487_VEBTi,F12: fun_bo851793448214054487_VEBTi,F22: fun_op5127434927081926045_VEBTi,F23: fun_op5127434927081926045_VEBTi] :
      ( ( Ti = Ti2 )
     => ( ! [A3: bool,B2: bool] : refine5565527176597971370_VEBTi(aa_boo772418440604470950_VEBTi(aa_boo6676949965781373448_VEBTi(F1,A3),B2),aa_boo772418440604470950_VEBTi(aa_boo6676949965781373448_VEBTi(F12,A3),B2))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F22,Info2),Deg2),TreeArray),Summary2),aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F23,Info2),Deg2),TreeArray),Summary2))
         => refine5565527176597971370_VEBTi(vEBT_c6028912655521741485_VEBTi(F22,F1,Ti),vEBT_c6028912655521741485_VEBTi(F23,F12,Ti2)) ) ) ) ).

% refines_case_VEBTi
tff(fact_1847_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: fun_bo7602104967519523767p_bool,F12: fun_bo7602104967519523767p_bool,F22: fun_op3392518339485081117p_bool,F23: fun_op3392518339485081117p_bool] :
      ( ( Ti = Ti2 )
     => ( ! [A3: bool,B2: bool] : refine4616567020171210338s_bool(aa_boo5261840290332707324p_bool(aa_boo5285575089431429040p_bool(F1,A3),B2),aa_boo5261840290332707324p_bool(aa_boo5285575089431429040p_bool(F12,A3),B2))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : refine4616567020171210338s_bool(aa_VEB101718416255554708p_bool(aa_arr628998674462452489p_bool(aa_nat5560262748439664842p_bool(aa_opt8498226628657232662p_bool(F22,Info2),Deg2),TreeArray),Summary2),aa_VEB101718416255554708p_bool(aa_arr628998674462452489p_bool(aa_nat5560262748439664842p_bool(aa_opt8498226628657232662p_bool(F23,Info2),Deg2),TreeArray),Summary2))
         => refine4616567020171210338s_bool(vEBT_c4579152981172362421p_bool(F22,F1,Ti),vEBT_c4579152981172362421p_bool(F23,F12,Ti2)) ) ) ) ).

% refines_case_VEBTi
tff(fact_1848_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(M),T))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H)),T)) ) ).

% TBOUNDD
tff(fact_1849_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_nat_bool(time_T8353473612707095248on_nat(M),T))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_option_nat(M,H)),T)) ) ).

% TBOUNDD
tff(fact_1850_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_bool,T: nat,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_nat_bool(time_TBOUND_bool(M),T))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_bool(M,H)),T)) ) ).

% TBOUNDD
tff(fact_1851_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,T: nat,H: heap_e7401611519738050253t_unit] :
      ( pp(aa_nat_bool(time_TBOUND_nat(M),T))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_nat(M,H)),T)) ) ).

% TBOUNDD
tff(fact_1852_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H2)),T))
     => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(M),T)) ) ).

% TBOUNDI
tff(fact_1853_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_option_nat(M,H2)),T))
     => pp(aa_nat_bool(time_T8353473612707095248on_nat(M),T)) ) ).

% TBOUNDI
tff(fact_1854_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_bool,T: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_bool(M,H2)),T))
     => pp(aa_nat_bool(time_TBOUND_bool(M),T)) ) ).

% TBOUNDI
tff(fact_1855_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,T: nat] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_nat(M,H2)),T))
     => pp(aa_nat_bool(time_TBOUND_nat(M),T)) ) ).

% TBOUNDI
tff(fact_1856_TBOUND__def,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(M),T))
    <=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H3)),T)) ) ).

% TBOUND_def
tff(fact_1857_TBOUND__def,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat] :
      ( pp(aa_nat_bool(time_T8353473612707095248on_nat(M),T))
    <=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_option_nat(M,H3)),T)) ) ).

% TBOUND_def
tff(fact_1858_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_bool,T: nat] :
      ( pp(aa_nat_bool(time_TBOUND_bool(M),T))
    <=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_bool(M,H3)),T)) ) ).

% TBOUND_def
tff(fact_1859_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_nat,T: nat] :
      ( pp(aa_nat_bool(time_TBOUND_nat(M),T))
    <=> ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_nat(M,H3)),T)) ) ).

% TBOUND_def
tff(fact_1860_TBOUND__mono,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,T: nat,T4: nat] :
      ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(C),T))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T4))
       => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(C),T4)) ) ) ).

% TBOUND_mono
tff(fact_1861_TBOUND__mono,axiom,
    ! [C: heap_T2636463487746394924on_nat,T: nat,T4: nat] :
      ( pp(aa_nat_bool(time_T8353473612707095248on_nat(C),T))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T4))
       => pp(aa_nat_bool(time_T8353473612707095248on_nat(C),T4)) ) ) ).

% TBOUND_mono
tff(fact_1862_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_bool,T: nat,T4: nat] :
      ( pp(aa_nat_bool(time_TBOUND_bool(C),T))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T4))
       => pp(aa_nat_bool(time_TBOUND_bool(C),T4)) ) ) ).

% TBOUND_mono
tff(fact_1863_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_nat,T: nat,T4: nat] :
      ( pp(aa_nat_bool(time_TBOUND_nat(C),T))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),T4))
       => pp(aa_nat_bool(time_TBOUND_nat(C),T4)) ) ) ).

% TBOUND_mono
tff(fact_1864_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( pp(aa_nat_bool(time_T8353473612707095248on_nat(C),T))
       => pp(aa_nat_bool(time_htt_option_nat(P,C,Q),T)) ) ) ).

% httI_TBOUND
tff(fact_1865_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,T: nat] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => ( pp(aa_nat_bool(time_TBOUND_bool(C),T))
       => pp(aa_nat_bool(time_htt_bool(P,C,Q),T)) ) ) ).

% httI_TBOUND
tff(fact_1866_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(C),T))
       => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,C,Q),T)) ) ) ).

% httI_TBOUND
tff(fact_1867_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( pp(aa_nat_bool(time_TBOUND_nat(C),T))
       => pp(aa_nat_bool(time_htt_nat(P,C,Q),T)) ) ) ).

% httI_TBOUND
tff(fact_1868_finite__transitivity__chain,axiom,
    ! [A2: set_VEBT_VEBT,R: fun_VE5877677516038742509T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [X: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,X),X))
       => ( ! [X: vEBT_VEBT,Y3: vEBT_VEBT,Z3: vEBT_VEBT] :
              ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,X),Y3))
             => ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,Y3),Z3))
               => pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,X),Z3)) ) )
         => ( ! [X: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
               => ? [Y4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y4),A2))
                    & pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(R,X),Y4)) ) )
           => ( A2 = bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1869_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: fun_se707272970763642173t_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( ! [X: set_nat] : ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X),X))
       => ( ! [X: set_nat,Y3: set_nat,Z3: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X),Y3))
             => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,Y3),Z3))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X),Z3)) ) )
         => ( ! [X: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
               => ? [Y4: set_nat] :
                    ( pp(aa_set_set_nat_bool(member_set_nat(Y4),A2))
                    & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(R,X),Y4)) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1870_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: fun_co1378794544173401837x_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X: complex] : ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X),X))
       => ( ! [X: complex,Y3: complex,Z3: complex] :
              ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X),Y3))
             => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y3),Z3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X),Z3)) ) )
         => ( ! [X: complex] :
                ( pp(aa_set_complex_bool(member_complex(X),A2))
               => ? [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),A2))
                    & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X),Y4)) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1871_finite__transitivity__chain,axiom,
    ! [A2: set_Code_integer,R: fun_Co8310303586393106575r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ! [X: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R,X),X))
       => ( ! [X: code_integer,Y3: code_integer,Z3: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R,X),Y3))
             => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R,Y3),Z3))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R,X),Z3)) ) )
         => ( ! [X: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),A2))
               => ? [Y4: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Y4),A2))
                    & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(R,X),Y4)) ) )
           => ( A2 = bot_bo3990330152332043303nteger ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1872_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: fun_re1255401033344146673l_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X),X))
       => ( ! [X: real,Y3: real,Z3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X),Y3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y3),Z3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X),Z3)) ) )
         => ( ! [X: real] :
                ( pp(aa_set_real_bool(member_real(X),A2))
               => ? [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),A2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X),Y4)) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1873_finite__transitivity__chain,axiom,
    ! [A2: set_bool,R: fun_bo1841576699660343469l_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ! [X: bool] : ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,X),X))
       => ( ! [X: bool,Y3: bool,Z3: bool] :
              ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,X),Y3))
             => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,Y3),Z3))
               => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,X),Z3)) ) )
         => ( ! [X: bool] :
                ( pp(aa_set_bool_bool(member_bool(X),A2))
               => ? [Y4: bool] :
                    ( pp(aa_set_bool_bool(member_bool(Y4),A2))
                    & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,X),Y4)) ) )
           => ( A2 = bot_bot_set_bool ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1874_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X),X))
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X),Z3)) ) )
         => ( ! [X: nat] :
                ( pp(aa_set_nat_bool(member_nat(X),A2))
               => ? [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),A2))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X),Y4)) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1875_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: fun_int_fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(R,X),X))
       => ( ! [X: int,Y3: int,Z3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(R,X),Y3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(R,Y3),Z3))
               => pp(aa_int_bool(aa_int_fun_int_bool(R,X),Z3)) ) )
         => ( ! [X: int] :
                ( pp(aa_set_int_bool(member_int(X),A2))
               => ? [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),A2))
                    & pp(aa_int_bool(aa_int_fun_int_bool(R,X),Y4)) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1876_arg__min__least,axiom,
    ! [S3: set_VEBT_VEBT,Y: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ( ( S3 != bot_bo8194388402131092736T_VEBT )
       => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,lattic6139528642216935859BT_rat(F,S3))),aa_VEBT_VEBT_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1877_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S3))),aa_complex_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1878_arg__min__least,axiom,
    ! [S3: set_Code_integer,Y: code_integer,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(F,lattic5439806495466278992er_rat(F,S3))),aa_Code_integer_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1879_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( S3 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(member_real(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S3))),aa_real_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1880_arg__min__least,axiom,
    ! [S3: set_bool,Y: bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ( ( S3 != bot_bot_set_bool )
       => ( pp(aa_set_bool_bool(member_bool(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,aa_set_bool_bool(lattic6202137068483785622ol_rat(F),S3))),aa_bool_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1881_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S3))),aa_nat_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1882_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( S3 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,aa_set_int_int(lattic7811156612396918303nt_rat(F),S3))),aa_int_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1883_arg__min__least,axiom,
    ! [S3: set_VEBT_VEBT,Y: vEBT_VEBT,F: fun_VEBT_VEBT_num] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ( ( S3 != bot_bo8194388402131092736T_VEBT )
       => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y),S3))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_VEBT_VEBT_num(F,lattic3331990488459210229BT_num(F,S3))),aa_VEBT_VEBT_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1884_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S3))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S3))),aa_complex_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1885_arg__min__least,axiom,
    ! [S3: set_Code_integer,Y: code_integer,F: fun_Code_integer_num] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Y),S3))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_Code_integer_num(F,lattic2632268341708553362er_num(F,S3))),aa_Code_integer_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1886_refines__replicate,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,F3: heap_T8145700208782473153_VEBTi,N: nat] :
      ( refine5565527176597971370_VEBTi(F,F3)
     => refine3700189196150522554_VEBTi(vEBT_V1859673955506687831_VEBTi(N,F),vEBT_V1859673955506687831_VEBTi(N,F3)) ) ).

% refines_replicate
tff(fact_1887_refines__replicate,axiom,
    ! [F: heap_Time_Heap_bool,F3: heap_Time_Heap_bool,N: nat] :
      ( refine4616567020171210338s_bool(F,F3)
     => refine3902108050975373032t_bool(vEBT_V6044891547498605301i_bool(N,F),vEBT_V6044891547498605301i_bool(N,F3)) ) ).

% refines_replicate
tff(fact_1888_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,C3: heap_T2636463487746394924on_nat] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( refine7594492741263601813on_nat(C3,C)
       => hoare_7629718768684598413on_nat(P,C3,Q) ) ) ).

% hoare_triple_refines
tff(fact_1889_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,C3: heap_Time_Heap_bool] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => ( refine4616567020171210338s_bool(C3,C)
       => hoare_2774773592644289642e_bool(P,C3,Q) ) ) ).

% hoare_triple_refines
tff(fact_1890_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,C3: heap_T8145700208782473153_VEBTi] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( refine5565527176597971370_VEBTi(C3,C)
       => hoare_1429296392585015714_VEBTi(P,C3,Q) ) ) ).

% hoare_triple_refines
tff(fact_1891_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,C3: heap_Time_Heap_nat] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( refine1365783493865988805es_nat(C3,C)
       => hoare_3067605981109127869le_nat(P,C3,Q) ) ) ).

% hoare_triple_refines
tff(fact_1892_vebt__maxt_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),X3))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( ( pp(B2)
                   => ( Y = aa_nat_option_nat(some_nat,one_one_nat) ) )
                  & ( ~ pp(B2)
                   => ( ( pp(A3)
                       => ( Y = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
                      & ( ~ pp(A3)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = aa_nat_option_nat(some_nat,Ma2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_1893_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)))) ) ).

% height_node
tff(fact_1894_vebt__mint_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),X3))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( ( pp(A3)
                   => ( Y = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
                  & ( ~ pp(A3)
                   => ( ( pp(B2)
                       => ( Y = aa_nat_option_nat(some_nat,one_one_nat) ) )
                      & ( ~ pp(B2)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = aa_nat_option_nat(some_nat,Mi2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_1895_subset__emptyI,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ! [X: vEBT_VEBT] : ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),bot_bo8194388402131092736T_VEBT)) ) ).

% subset_emptyI
tff(fact_1896_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X: set_nat] : ~ pp(aa_set_set_nat_bool(member_set_nat(X),A2))
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),bot_bot_set_set_nat)) ) ).

% subset_emptyI
tff(fact_1897_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X: real] : ~ pp(aa_set_real_bool(member_real(X),A2))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),bot_bot_set_real)) ) ).

% subset_emptyI
tff(fact_1898_subset__emptyI,axiom,
    ! [A2: set_bool] :
      ( ! [X: bool] : ~ pp(aa_set_bool_bool(member_bool(X),A2))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),bot_bot_set_bool)) ) ).

% subset_emptyI
tff(fact_1899_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X: nat] : ~ pp(aa_set_nat_bool(member_nat(X),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),bot_bot_set_nat)) ) ).

% subset_emptyI
tff(fact_1900_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X: int] : ~ pp(aa_set_int_bool(member_int(X),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),bot_bot_set_int)) ) ).

% subset_emptyI
tff(fact_1901_complete__interval,axiom,
    ! [A: real,B: real,P: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(P,A))
       => ( ~ pp(aa_real_bool(P,B))
         => ? [C4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C4),B))
              & ! [X5: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X5))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),C4)) )
                 => pp(aa_real_bool(P,X5)) )
              & ! [D3: real] :
                  ( ! [X: real] :
                      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
                        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),D3)) )
                     => pp(aa_real_bool(P,X)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D3),C4)) ) ) ) ) ) ).

% complete_interval
tff(fact_1902_complete__interval,axiom,
    ! [A: nat,B: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(P,A))
       => ( ~ pp(aa_nat_bool(P,B))
         => ? [C4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C4),B))
              & ! [X5: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X5))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),C4)) )
                 => pp(aa_nat_bool(P,X5)) )
              & ! [D3: nat] :
                  ( ! [X: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),D3)) )
                     => pp(aa_nat_bool(P,X)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D3),C4)) ) ) ) ) ) ).

% complete_interval
tff(fact_1903_complete__interval,axiom,
    ! [A: int,B: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(P,A))
       => ( ~ pp(aa_int_bool(P,B))
         => ? [C4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C4))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C4),B))
              & ! [X5: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X5))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),C4)) )
                 => pp(aa_int_bool(P,X5)) )
              & ! [D3: int] :
                  ( ! [X: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),D3)) )
                     => pp(aa_int_bool(P,X)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D3),C4)) ) ) ) ) ) ).

% complete_interval
tff(fact_1904_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A5: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B5),A5))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1905_verit__comp__simplify1_I3_J,axiom,
    ! [B5: rat,A5: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B5),A5))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1906_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A5: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B5),A5))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1907_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B5),A5))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1908_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),A5))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_1909_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X5))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),T)) ) ).

% pinf(6)
tff(fact_1910_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X5))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),T)) ) ).

% pinf(6)
tff(fact_1911_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X5))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),T)) ) ).

% pinf(6)
tff(fact_1912_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X5))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),T)) ) ).

% pinf(6)
tff(fact_1913_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X5))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T)) ) ).

% pinf(6)
tff(fact_1914_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X5))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X5)) ) ).

% pinf(8)
tff(fact_1915_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X5))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X5)) ) ).

% pinf(8)
tff(fact_1916_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X5))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X5)) ) ).

% pinf(8)
tff(fact_1917_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X5)) ) ).

% pinf(8)
tff(fact_1918_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X5))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5)) ) ).

% pinf(8)
tff(fact_1919_vebt__memberi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X3: nat,T: vEBT_VEBT] : refine4616567020171210338s_bool(vEBT_vebt_memberi(Ti,X3),vEBT_V854960066525838166emberi(T,Ti,X3)) ).

% vebt_memberi_refines
tff(fact_1920_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = zero_zero_nat ).

% VEBT_internal.height.simps(1)
tff(fact_1921_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1922_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1923_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1924_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1925_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1926_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% verit_la_disequality
tff(fact_1927_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% verit_la_disequality
tff(fact_1928_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% verit_la_disequality
tff(fact_1929_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% verit_la_disequality
tff(fact_1930_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X5)) ) ).

% minf(7)
tff(fact_1931_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X5)) ) ).

% minf(7)
tff(fact_1932_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X5)) ) ).

% minf(7)
tff(fact_1933_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X5)) ) ).

% minf(7)
tff(fact_1934_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X5)) ) ).

% minf(7)
tff(fact_1935_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),T)) ) ).

% minf(5)
tff(fact_1936_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),T)) ) ).

% minf(5)
tff(fact_1937_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),T)) ) ).

% minf(5)
tff(fact_1938_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),T)) ) ).

% minf(5)
tff(fact_1939_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),T)) ) ).

% minf(5)
tff(fact_1940_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z3))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_1941_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z3))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_1942_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z3))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_1943_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z3))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_1944_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z3))
     => ( X5 != T ) ) ).

% minf(4)
tff(fact_1945_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z3))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_1946_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z3))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_1947_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z3))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_1948_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z3))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_1949_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z3))
     => ( X5 != T ) ) ).

% minf(3)
tff(fact_1950_minf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z4))
         => ( pp(aa_real_bool(P,X))
          <=> pp(aa_real_bool(P2,X)) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z4))
           => ( pp(aa_real_bool(Q,X))
            <=> pp(aa_real_bool(Q2,X)) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z3))
           => ( ( pp(aa_real_bool(P,X5))
                | pp(aa_real_bool(Q,X5)) )
            <=> ( pp(aa_real_bool(P2,X5))
                | pp(aa_real_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_1951_minf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z4))
         => ( pp(aa_rat_bool(P,X))
          <=> pp(aa_rat_bool(P2,X)) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z4))
           => ( pp(aa_rat_bool(Q,X))
            <=> pp(aa_rat_bool(Q2,X)) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z3))
           => ( ( pp(aa_rat_bool(P,X5))
                | pp(aa_rat_bool(Q,X5)) )
            <=> ( pp(aa_rat_bool(P2,X5))
                | pp(aa_rat_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_1952_minf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z4))
         => ( pp(aa_num_bool(P,X))
          <=> pp(aa_num_bool(P2,X)) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z4))
           => ( pp(aa_num_bool(Q,X))
            <=> pp(aa_num_bool(Q2,X)) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z3))
           => ( ( pp(aa_num_bool(P,X5))
                | pp(aa_num_bool(Q,X5)) )
            <=> ( pp(aa_num_bool(P2,X5))
                | pp(aa_num_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_1953_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z4))
         => ( pp(aa_nat_bool(P,X))
          <=> pp(aa_nat_bool(P2,X)) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z4))
           => ( pp(aa_nat_bool(Q,X))
            <=> pp(aa_nat_bool(Q2,X)) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z3))
           => ( ( pp(aa_nat_bool(P,X5))
                | pp(aa_nat_bool(Q,X5)) )
            <=> ( pp(aa_nat_bool(P2,X5))
                | pp(aa_nat_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_1954_minf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z4))
         => ( pp(aa_int_bool(P,X))
          <=> pp(aa_int_bool(P2,X)) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z4))
           => ( pp(aa_int_bool(Q,X))
            <=> pp(aa_int_bool(Q2,X)) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z3))
           => ( ( pp(aa_int_bool(P,X5))
                | pp(aa_int_bool(Q,X5)) )
            <=> ( pp(aa_int_bool(P2,X5))
                | pp(aa_int_bool(Q2,X5)) ) ) ) ) ) ).

% minf(2)
tff(fact_1955_minf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z4))
         => ( pp(aa_real_bool(P,X))
          <=> pp(aa_real_bool(P2,X)) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z4))
           => ( pp(aa_real_bool(Q,X))
            <=> pp(aa_real_bool(Q2,X)) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z3))
           => ( ( pp(aa_real_bool(P,X5))
                & pp(aa_real_bool(Q,X5)) )
            <=> ( pp(aa_real_bool(P2,X5))
                & pp(aa_real_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_1956_minf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z4))
         => ( pp(aa_rat_bool(P,X))
          <=> pp(aa_rat_bool(P2,X)) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z4))
           => ( pp(aa_rat_bool(Q,X))
            <=> pp(aa_rat_bool(Q2,X)) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z3))
           => ( ( pp(aa_rat_bool(P,X5))
                & pp(aa_rat_bool(Q,X5)) )
            <=> ( pp(aa_rat_bool(P2,X5))
                & pp(aa_rat_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_1957_minf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z4))
         => ( pp(aa_num_bool(P,X))
          <=> pp(aa_num_bool(P2,X)) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z4))
           => ( pp(aa_num_bool(Q,X))
            <=> pp(aa_num_bool(Q2,X)) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z3))
           => ( ( pp(aa_num_bool(P,X5))
                & pp(aa_num_bool(Q,X5)) )
            <=> ( pp(aa_num_bool(P2,X5))
                & pp(aa_num_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_1958_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z4))
         => ( pp(aa_nat_bool(P,X))
          <=> pp(aa_nat_bool(P2,X)) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z4))
           => ( pp(aa_nat_bool(Q,X))
            <=> pp(aa_nat_bool(Q2,X)) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z3))
           => ( ( pp(aa_nat_bool(P,X5))
                & pp(aa_nat_bool(Q,X5)) )
            <=> ( pp(aa_nat_bool(P2,X5))
                & pp(aa_nat_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_1959_minf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z4))
         => ( pp(aa_int_bool(P,X))
          <=> pp(aa_int_bool(P2,X)) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z4))
           => ( pp(aa_int_bool(Q,X))
            <=> pp(aa_int_bool(Q2,X)) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z3))
           => ( ( pp(aa_int_bool(P,X5))
                & pp(aa_int_bool(Q,X5)) )
            <=> ( pp(aa_int_bool(P2,X5))
                & pp(aa_int_bool(Q2,X5)) ) ) ) ) ) ).

% minf(1)
tff(fact_1960_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X5))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X5)) ) ).

% pinf(7)
tff(fact_1961_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X5))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X5)) ) ).

% pinf(7)
tff(fact_1962_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X5))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X5)) ) ).

% pinf(7)
tff(fact_1963_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X5)) ) ).

% pinf(7)
tff(fact_1964_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X5))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X5)) ) ).

% pinf(7)
tff(fact_1965_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X5))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),T)) ) ).

% pinf(5)
tff(fact_1966_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X5))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),T)) ) ).

% pinf(5)
tff(fact_1967_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X5))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),T)) ) ).

% pinf(5)
tff(fact_1968_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X5))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),T)) ) ).

% pinf(5)
tff(fact_1969_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X5))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),T)) ) ).

% pinf(5)
tff(fact_1970_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_1971_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_1972_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_1973_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_1974_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(4)
tff(fact_1975_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_1976_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_1977_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_1978_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_1979_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X5))
     => ( X5 != T ) ) ).

% pinf(3)
tff(fact_1980_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X))
         => ( pp(aa_real_bool(P,X))
          <=> pp(aa_real_bool(P2,X)) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X))
           => ( pp(aa_real_bool(Q,X))
            <=> pp(aa_real_bool(Q2,X)) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X5))
           => ( ( pp(aa_real_bool(P,X5))
                | pp(aa_real_bool(Q,X5)) )
            <=> ( pp(aa_real_bool(P2,X5))
                | pp(aa_real_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1981_pinf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X))
         => ( pp(aa_rat_bool(P,X))
          <=> pp(aa_rat_bool(P2,X)) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X))
           => ( pp(aa_rat_bool(Q,X))
            <=> pp(aa_rat_bool(Q2,X)) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X5))
           => ( ( pp(aa_rat_bool(P,X5))
                | pp(aa_rat_bool(Q,X5)) )
            <=> ( pp(aa_rat_bool(P2,X5))
                | pp(aa_rat_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1982_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X))
         => ( pp(aa_num_bool(P,X))
          <=> pp(aa_num_bool(P2,X)) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X))
           => ( pp(aa_num_bool(Q,X))
            <=> pp(aa_num_bool(Q2,X)) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X5))
           => ( ( pp(aa_num_bool(P,X5))
                | pp(aa_num_bool(Q,X5)) )
            <=> ( pp(aa_num_bool(P2,X5))
                | pp(aa_num_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1983_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X))
         => ( pp(aa_nat_bool(P,X))
          <=> pp(aa_nat_bool(P2,X)) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X))
           => ( pp(aa_nat_bool(Q,X))
            <=> pp(aa_nat_bool(Q2,X)) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X5))
           => ( ( pp(aa_nat_bool(P,X5))
                | pp(aa_nat_bool(Q,X5)) )
            <=> ( pp(aa_nat_bool(P2,X5))
                | pp(aa_nat_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1984_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X))
         => ( pp(aa_int_bool(P,X))
          <=> pp(aa_int_bool(P2,X)) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X))
           => ( pp(aa_int_bool(Q,X))
            <=> pp(aa_int_bool(Q2,X)) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X5))
           => ( ( pp(aa_int_bool(P,X5))
                | pp(aa_int_bool(Q,X5)) )
            <=> ( pp(aa_int_bool(P2,X5))
                | pp(aa_int_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1985_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q2: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X))
         => ( pp(aa_real_bool(P,X))
          <=> pp(aa_real_bool(P2,X)) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X))
           => ( pp(aa_real_bool(Q,X))
            <=> pp(aa_real_bool(Q2,X)) ) )
       => ? [Z3: real] :
          ! [X5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X5))
           => ( ( pp(aa_real_bool(P,X5))
                & pp(aa_real_bool(Q,X5)) )
            <=> ( pp(aa_real_bool(P2,X5))
                & pp(aa_real_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1986_pinf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q2: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X))
         => ( pp(aa_rat_bool(P,X))
          <=> pp(aa_rat_bool(P2,X)) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X))
           => ( pp(aa_rat_bool(Q,X))
            <=> pp(aa_rat_bool(Q2,X)) ) )
       => ? [Z3: rat] :
          ! [X5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X5))
           => ( ( pp(aa_rat_bool(P,X5))
                & pp(aa_rat_bool(Q,X5)) )
            <=> ( pp(aa_rat_bool(P2,X5))
                & pp(aa_rat_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1987_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X))
         => ( pp(aa_num_bool(P,X))
          <=> pp(aa_num_bool(P2,X)) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X))
           => ( pp(aa_num_bool(Q,X))
            <=> pp(aa_num_bool(Q2,X)) ) )
       => ? [Z3: num] :
          ! [X5: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X5))
           => ( ( pp(aa_num_bool(P,X5))
                & pp(aa_num_bool(Q,X5)) )
            <=> ( pp(aa_num_bool(P2,X5))
                & pp(aa_num_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1988_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X))
         => ( pp(aa_nat_bool(P,X))
          <=> pp(aa_nat_bool(P2,X)) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X))
           => ( pp(aa_nat_bool(Q,X))
            <=> pp(aa_nat_bool(Q2,X)) ) )
       => ? [Z3: nat] :
          ! [X5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X5))
           => ( ( pp(aa_nat_bool(P,X5))
                & pp(aa_nat_bool(Q,X5)) )
            <=> ( pp(aa_nat_bool(P2,X5))
                & pp(aa_nat_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1989_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X))
         => ( pp(aa_int_bool(P,X))
          <=> pp(aa_int_bool(P2,X)) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X))
           => ( pp(aa_int_bool(Q,X))
            <=> pp(aa_int_bool(Q2,X)) ) )
       => ? [Z3: int] :
          ! [X5: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X5))
           => ( ( pp(aa_int_bool(P,X5))
                & pp(aa_int_bool(Q,X5)) )
            <=> ( pp(aa_int_bool(P2,X5))
                & pp(aa_int_bool(Q2,X5)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1990_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1991_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1992_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1993_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1994_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1995_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B2))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),A)) ) ).

% ex_gt_or_lt
tff(fact_1996_prop__restrict,axiom,
    ! [X3: vEBT_VEBT,Z5: set_VEBT_VEBT,X6: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),Z5))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(Z5),collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bg(X6),P))))
       => pp(aa_VEBT_VEBT_bool(P,X3)) ) ) ).

% prop_restrict
tff(fact_1997_prop__restrict,axiom,
    ! [X3: real,Z5: set_real,X6: set_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(X3),Z5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(Z5),collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bh(X6),P))))
       => pp(aa_real_bool(P,X3)) ) ) ).

% prop_restrict
tff(fact_1998_prop__restrict,axiom,
    ! [X3: complex,Z5: set_complex,X6: set_complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(member_complex(X3),Z5))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(Z5),collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_bi(X6),P))))
       => pp(aa_complex_bool(P,X3)) ) ) ).

% prop_restrict
tff(fact_1999_prop__restrict,axiom,
    ! [X3: list_nat,Z5: set_list_nat,X6: set_list_nat,P: fun_list_nat_bool] :
      ( pp(aa_set_list_nat_bool(member_list_nat(X3),Z5))
     => ( ord_le6045566169113846134st_nat(Z5,collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bj(X6),P)))
       => pp(aa_list_nat_bool(P,X3)) ) ) ).

% prop_restrict
tff(fact_2000_prop__restrict,axiom,
    ! [X3: set_nat,Z5: set_set_nat,X6: set_set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),Z5))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(Z5),collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bk(X6),P))))
       => pp(aa_set_nat_bool(P,X3)) ) ) ).

% prop_restrict
tff(fact_2001_prop__restrict,axiom,
    ! [X3: nat,Z5: set_nat,X6: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(X3),Z5))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z5),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bl(X6),P))))
       => pp(aa_nat_bool(P,X3)) ) ) ).

% prop_restrict
tff(fact_2002_prop__restrict,axiom,
    ! [X3: int,Z5: set_int,X6: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(X3),Z5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Z5),collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bm(X6),P))))
       => pp(aa_int_bool(P,X3)) ) ) ).

% prop_restrict
tff(fact_2003_Collect__restrict,axiom,
    ! [X6: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bg(X6),P))),X6)) ).

% Collect_restrict
tff(fact_2004_Collect__restrict,axiom,
    ! [X6: set_real,P: fun_real_bool] : pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bh(X6),P))),X6)) ).

% Collect_restrict
tff(fact_2005_Collect__restrict,axiom,
    ! [X6: set_complex,P: fun_complex_bool] : pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_bi(X6),P))),X6)) ).

% Collect_restrict
tff(fact_2006_Collect__restrict,axiom,
    ! [X6: set_list_nat,P: fun_list_nat_bool] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bj(X6),P)),X6) ).

% Collect_restrict
tff(fact_2007_Collect__restrict,axiom,
    ! [X6: set_set_nat,P: fun_set_nat_bool] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bk(X6),P))),X6)) ).

% Collect_restrict
tff(fact_2008_Collect__restrict,axiom,
    ! [X6: set_nat,P: fun_nat_bool] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bl(X6),P))),X6)) ).

% Collect_restrict
tff(fact_2009_Collect__restrict,axiom,
    ! [X6: set_int,P: fun_int_bool] : pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bm(X6),P))),X6)) ).

% Collect_restrict
tff(fact_2010_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X5)) ) ).

% minf(8)
tff(fact_2011_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X5)) ) ).

% minf(8)
tff(fact_2012_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X5)) ) ).

% minf(8)
tff(fact_2013_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X5)) ) ).

% minf(8)
tff(fact_2014_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5)) ) ).

% minf(8)
tff(fact_2015_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X5: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),Z3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),T)) ) ).

% minf(6)
tff(fact_2016_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X5: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),Z3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),T)) ) ).

% minf(6)
tff(fact_2017_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X5: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),Z3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),T)) ) ).

% minf(6)
tff(fact_2018_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Z3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),T)) ) ).

% minf(6)
tff(fact_2019_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),Z3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T)) ) ).

% minf(6)
tff(fact_2020_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,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),X3))
       => ( ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
           => ( ( Y = one_one_nat )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse))) ) )
         => ( ! [Uv2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2) )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2))) ) )
           => ( ! [Uu2: bool] :
                  ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue))) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Uz2),Va3,Vb2),Vc2) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_2021_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,T),X3)),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_2022_time__replicate,axiom,
    ! [X3: heap_T8145700208782473153_VEBTi,C: nat,N: nat,H: heap_e7401611519738050253t_unit] :
      ( ! [H2: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(X3,H2)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_t3534373299052942712_VEBTi(vEBT_V1859673955506687831_VEBTi(N,X3),H)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N)))) ) ).

% time_replicate
tff(fact_2023_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : pp(aa_nat_bool(time_TBOUND_bool(vEBT_VEBT_minNulli(T)),one_one_nat)) ).

% TBOUND_minNulli
tff(fact_2024_TBOUND__replicate,axiom,
    ! [X3: heap_T8145700208782473153_VEBTi,C: nat,N: nat] :
      ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(X3),C))
     => pp(aa_nat_bool(time_T8149879359713347829_VEBTi(vEBT_V1859673955506687831_VEBTi(N,X3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N)))) ) ).

% TBOUND_replicate
tff(fact_2025_TBOUND__replicate,axiom,
    ! [X3: heap_T2636463487746394924on_nat,C: nat,N: nat] :
      ( pp(aa_nat_bool(time_T8353473612707095248on_nat(X3),C))
     => pp(aa_nat_bool(time_T3808005469503390304on_nat(vEBT_V792416675989592002on_nat(N,X3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N)))) ) ).

% TBOUND_replicate
tff(fact_2026_TBOUND__replicate,axiom,
    ! [X3: heap_Time_Heap_bool,C: nat,N: nat] :
      ( pp(aa_nat_bool(time_TBOUND_bool(X3),C))
     => pp(aa_nat_bool(time_T5840735555483777133t_bool(vEBT_V6044891547498605301i_bool(N,X3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N)))) ) ).

% TBOUND_replicate
tff(fact_2027_TBOUND__replicate,axiom,
    ! [X3: heap_Time_Heap_nat,C: nat,N: nat] :
      ( pp(aa_nat_bool(time_TBOUND_nat(X3),C))
     => pp(aa_nat_bool(time_TBOUND_list_nat(vEBT_V7726092123322077554ei_nat(N,X3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N)))) ) ).

% TBOUND_replicate
tff(fact_2028_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2029_accp__subset,axiom,
    ! [R1: fun_Pr817965127378462947t_bool,R22: fun_Pr817965127378462947t_bool] :
      ( ord_le5934820504122867843t_bool(R1,R22)
     => ord_le7893316316344127773t_bool(accp_P2887432264394892906BT_nat(R22),accp_P2887432264394892906BT_nat(R1)) ) ).

% accp_subset
tff(fact_2030_accp__subset,axiom,
    ! [R1: fun_Pr248683899324676269t_bool,R22: fun_Pr248683899324676269t_bool] :
      ( ord_le2087689725136068685t_bool(R1,R22)
     => ord_le2468614981408567801t_bool(accp_P4275260045618599050at_nat(R22),accp_P4275260045618599050at_nat(R1)) ) ).

% accp_subset
tff(fact_2031_accp__subset,axiom,
    ! [R1: fun_Pr3099820731930151597t_bool,R22: fun_Pr3099820731930151597t_bool] :
      ( ord_le4938826557741544013t_bool(R1,R22)
     => ord_le4737936599360737601t_bool(accp_P1096762738010456898nt_int(R22),accp_P1096762738010456898nt_int(R1)) ) ).

% accp_subset
tff(fact_2032_accp__subset,axiom,
    ! [R1: fun_li8598321315282354825t_bool,R22: fun_li8598321315282354825t_bool] :
      ( ord_le4434192926011882025t_bool(R1,R22)
     => ord_le2936377143570524336t_bool(accp_list_nat(R22),accp_list_nat(R1)) ) ).

% accp_subset
tff(fact_2033_accp__subset,axiom,
    ! [R1: fun_nat_fun_nat_bool,R22: fun_nat_fun_nat_bool] :
      ( ord_le4899916226251588169t_bool(R1,R22)
     => ord_le8840051076831633728t_bool(accp_nat(R22),accp_nat(R1)) ) ).

% accp_subset
tff(fact_2034_accp__subset__induct,axiom,
    ! [D4: fun_Pr2834072668759824765t_bool,R: fun_Pr817965127378462947t_bool,X3: produc9072475918466114483BT_nat,P: fun_Pr2834072668759824765t_bool] :
      ( ord_le7893316316344127773t_bool(D4,accp_P2887432264394892906BT_nat(R))
     => ( ! [X: produc9072475918466114483BT_nat,Z3: produc9072475918466114483BT_nat] :
            ( pp(aa_Pro1009365227942981478t_bool(D4,X))
           => ( pp(aa_Pro1009365227942981478t_bool(aa_Pro263205025495559628t_bool(R,Z3),X))
             => pp(aa_Pro1009365227942981478t_bool(D4,Z3)) ) )
       => ( pp(aa_Pro1009365227942981478t_bool(D4,X3))
         => ( ! [X: produc9072475918466114483BT_nat] :
                ( pp(aa_Pro1009365227942981478t_bool(D4,X))
               => ( ! [Z4: produc9072475918466114483BT_nat] :
                      ( pp(aa_Pro1009365227942981478t_bool(aa_Pro263205025495559628t_bool(R,Z4),X))
                     => pp(aa_Pro1009365227942981478t_bool(P,Z4)) )
                 => pp(aa_Pro1009365227942981478t_bool(P,X)) ) )
           => pp(aa_Pro1009365227942981478t_bool(P,X3)) ) ) ) ) ).

% accp_subset_induct
tff(fact_2035_accp__subset__induct,axiom,
    ! [D4: fun_Pr7757163693017731481t_bool,R: fun_Pr248683899324676269t_bool,X3: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool] :
      ( ord_le2468614981408567801t_bool(D4,accp_P4275260045618599050at_nat(R))
     => ( ! [X: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( pp(aa_Pro2232559814647311250t_bool(D4,X))
           => ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,Z3),X))
             => pp(aa_Pro2232559814647311250t_bool(D4,Z3)) ) )
       => ( pp(aa_Pro2232559814647311250t_bool(D4,X3))
         => ( ! [X: product_prod_nat_nat] :
                ( pp(aa_Pro2232559814647311250t_bool(D4,X))
               => ( ! [Z4: product_prod_nat_nat] :
                      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,Z4),X))
                     => pp(aa_Pro2232559814647311250t_bool(P,Z4)) )
                 => pp(aa_Pro2232559814647311250t_bool(P,X)) ) )
           => pp(aa_Pro2232559814647311250t_bool(P,X3)) ) ) ) ) ).

% accp_subset_induct
tff(fact_2036_accp__subset__induct,axiom,
    ! [D4: fun_Pr803113274115125473t_bool,R: fun_Pr3099820731930151597t_bool,X3: product_prod_int_int,P: fun_Pr803113274115125473t_bool] :
      ( ord_le4737936599360737601t_bool(D4,accp_P1096762738010456898nt_int(R))
     => ( ! [X: product_prod_int_int,Z3: product_prod_int_int] :
            ( pp(aa_Pro4501881432599481050t_bool(D4,X))
           => ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(R,Z3),X))
             => pp(aa_Pro4501881432599481050t_bool(D4,Z3)) ) )
       => ( pp(aa_Pro4501881432599481050t_bool(D4,X3))
         => ( ! [X: product_prod_int_int] :
                ( pp(aa_Pro4501881432599481050t_bool(D4,X))
               => ( ! [Z4: product_prod_int_int] :
                      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(R,Z4),X))
                     => pp(aa_Pro4501881432599481050t_bool(P,Z4)) )
                 => pp(aa_Pro4501881432599481050t_bool(P,X)) ) )
           => pp(aa_Pro4501881432599481050t_bool(P,X3)) ) ) ) ) ).

% accp_subset_induct
tff(fact_2037_accp__subset__induct,axiom,
    ! [D4: fun_list_nat_bool,R: fun_li8598321315282354825t_bool,X3: list_nat,P: fun_list_nat_bool] :
      ( ord_le2936377143570524336t_bool(D4,accp_list_nat(R))
     => ( ! [X: list_nat,Z3: list_nat] :
            ( pp(aa_list_nat_bool(D4,X))
           => ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(R,Z3),X))
             => pp(aa_list_nat_bool(D4,Z3)) ) )
       => ( pp(aa_list_nat_bool(D4,X3))
         => ( ! [X: list_nat] :
                ( pp(aa_list_nat_bool(D4,X))
               => ( ! [Z4: list_nat] :
                      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(R,Z4),X))
                     => pp(aa_list_nat_bool(P,Z4)) )
                 => pp(aa_list_nat_bool(P,X)) ) )
           => pp(aa_list_nat_bool(P,X3)) ) ) ) ) ).

% accp_subset_induct
tff(fact_2038_accp__subset__induct,axiom,
    ! [D4: fun_nat_bool,R: fun_nat_fun_nat_bool,X3: nat,P: fun_nat_bool] :
      ( ord_le8840051076831633728t_bool(D4,accp_nat(R))
     => ( ! [X: nat,Z3: nat] :
            ( pp(aa_nat_bool(D4,X))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Z3),X))
             => pp(aa_nat_bool(D4,Z3)) ) )
       => ( pp(aa_nat_bool(D4,X3))
         => ( ! [X: nat] :
                ( pp(aa_nat_bool(D4,X))
               => ( ! [Z4: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Z4),X))
                     => pp(aa_nat_bool(P,Z4)) )
                 => pp(aa_nat_bool(P,X)) ) )
           => pp(aa_nat_bool(P,X3)) ) ) ) ) ).

% accp_subset_induct
tff(fact_2039_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D22))
       => ? [E: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D22)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2040_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D1))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D22))
       => ? [E: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D22)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2041_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,T) = none_nat ) ) ).

% minNullmin
tff(fact_2042_even__odd__cases,axiom,
    ! [X3: nat] :
      ( ! [N2: nat] : X3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : X3 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_2043_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X3: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X3)) ) ).

% min_Null_member
tff(fact_2044_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,T) = none_nat )
     => pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).

% minminNull
tff(fact_2045_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2046_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2047_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2048_add__left__cancel,axiom,
    ! [A: int,B: int,C: 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),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2049_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2050_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2051_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2052_add__right__cancel,axiom,
    ! [B: int,A: int,C: 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,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2053_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : pp(aa_nat_bool(time_htt_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_VEBT_minNulli(Ti),aa_VEB1972364096475303967l_assn(aTP_Lamm_eo(T),Ti)),one_one_nat)) ).

% minNrulli_ruleT
tff(fact_2054_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_2055_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_2056_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_2057_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_2058_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_2059_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_2060_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_2061_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_2062_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_2063_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_2064_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_2065_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_2066_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_2067_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_2068_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_2069_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_2070_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_2071_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_2072_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_2073_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_2074_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_2075_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_2076_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_2077_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_2078_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_2079_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_2080_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_2081_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_2082_add__eq__0__iff__both__eq__0,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) = zero_zero_nat )
    <=> ( ( X3 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2083_zero__eq__add__iff__both__eq__0,axiom,
    ! [X3: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) )
    <=> ( ( X3 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2084_add__0,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),A) = A ).

% add_0
tff(fact_2085_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_2086_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_2087_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_2088_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_2089_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_left
tff(fact_2090_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_left
tff(fact_2091_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_left
tff(fact_2092_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_left
tff(fact_2093_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_right
tff(fact_2094_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_right
tff(fact_2095_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_right
tff(fact_2096_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_right
tff(fact_2097_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_left
tff(fact_2098_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_left
tff(fact_2099_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_left
tff(fact_2100_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_left
tff(fact_2101_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_right
tff(fact_2102_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_right
tff(fact_2103_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_right
tff(fact_2104_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_right
tff(fact_2105_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc_right
tff(fact_2106_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_2107_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_2108_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_2109_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : hoare_2774773592644289642e_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_VEBT_minNulli(Ti),aa_VEB1972364096475303967l_assn(aTP_Lamm_eo(T),Ti)) ).

% minNulli_rule
tff(fact_2110_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_2111_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel1
tff(fact_2112_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel1
tff(fact_2113_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_2114_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_2115_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel2
tff(fact_2116_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel2
tff(fact_2117_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_2118_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_2119_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel1
tff(fact_2120_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel1
tff(fact_2121_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel1
tff(fact_2122_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel1
tff(fact_2123_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel2
tff(fact_2124_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel2
tff(fact_2125_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel2
tff(fact_2126_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel2
tff(fact_2127_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2128_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2129_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2130_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2131_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2132_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2133_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel1
tff(fact_2134_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel1
tff(fact_2135_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_2136_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_2137_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel2
tff(fact_2138_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel2
tff(fact_2139_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_2140_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_2141_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel1
tff(fact_2142_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel1
tff(fact_2143_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel1
tff(fact_2144_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel1
tff(fact_2145_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel2
tff(fact_2146_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel2
tff(fact_2147_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel2
tff(fact_2148_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel2
tff(fact_2149_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2150_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2151_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2152_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2153_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2154_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2155_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_2156_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc_right
tff(fact_2157_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X3: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(T,Ti,X3)),one_one_nat)) ) ).

% TBOUND_minNull
tff(fact_2158_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_2159_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2160_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_2161_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_2162_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2163_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2164_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2165_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),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_2166_group__cancel_Oadd2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2167_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2168_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2169_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3 = 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),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2170_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_2171_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.assoc
tff(fact_2172_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_2173_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_2174_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2175_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2176_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: 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),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2177_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2178_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2179_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: 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,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2180_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_2181_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_2182_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_2183_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_2184_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2185_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2186_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2187_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: 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),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2188_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2189_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2190_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2191_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: 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),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2192_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2193_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2194_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2195_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: 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,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2196_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2197_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_2198_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2199_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_2200_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_2201_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_2202_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_2203_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_2204_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_2205_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_2206_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_2207_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_2208_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_2209_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_2210_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_2211_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_2212_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_2213_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_2214_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_2215_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2216_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2217_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2218_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2219_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2220_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2221_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2222_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2223_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2224_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2225_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2226_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2227_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_mono
tff(fact_2228_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_mono
tff(fact_2229_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_mono
tff(fact_2230_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_mono
tff(fact_2231_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_left_mono
tff(fact_2232_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% add_left_mono
tff(fact_2233_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_left_mono
tff(fact_2234_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_left_mono
tff(fact_2235_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ~ ! [C4: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% less_eqE
tff(fact_2236_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_right_mono
tff(fact_2237_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))) ) ).

% add_right_mono
tff(fact_2238_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_right_mono
tff(fact_2239_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_right_mono
tff(fact_2240_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ? [C5: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C5) ) ).

% le_iff_add
tff(fact_2241_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_left
tff(fact_2242_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_2243_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_2244_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_left
tff(fact_2245_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_right
tff(fact_2246_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_2247_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_2248_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_right
tff(fact_2249_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2250_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2251_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2252_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2253_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2254_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2255_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2256_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2257_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2258_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2259_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2260_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2261_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_strict_mono
tff(fact_2262_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_strict_mono
tff(fact_2263_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_strict_mono
tff(fact_2264_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_strict_mono
tff(fact_2265_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_strict_left_mono
tff(fact_2266_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% add_strict_left_mono
tff(fact_2267_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_strict_left_mono
tff(fact_2268_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_strict_left_mono
tff(fact_2269_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_strict_right_mono
tff(fact_2270_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))) ) ).

% add_strict_right_mono
tff(fact_2271_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_strict_right_mono
tff(fact_2272_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_strict_right_mono
tff(fact_2273_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_left
tff(fact_2274_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_2275_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_2276_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_left
tff(fact_2277_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_right
tff(fact_2278_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_2279_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_2280_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_right
tff(fact_2281_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2282_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2283_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2284_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2285_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2286_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2287_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_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,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2288_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2289_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2290_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2291_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2292_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2293_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2294_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_2295_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% distrib_left
tff(fact_2296_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_2297_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_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,A),C)) ).

% distrib_left
tff(fact_2298_distrib__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)) ).

% distrib_left
tff(fact_2299_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_2300_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% distrib_right
tff(fact_2301_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_2302_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_2303_distrib__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) ).

% distrib_right
tff(fact_2304_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2305_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2306_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2307_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2308_combine__common__factor,axiom,
    ! [A: complex,E2: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),E2)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),E2)),C)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2309_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_2310_add__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc
tff(fact_2311_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_2312_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_2313_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_2314_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% add_lessD1
tff(fact_2315_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_less_mono
tff(fact_2316_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).

% not_add_less1
tff(fact_2317_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).

% not_add_less2
tff(fact_2318_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_less_mono1
tff(fact_2319_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_less_add1
tff(fact_2320_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_less_add2
tff(fact_2321_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% less_add_eq_less
tff(fact_2322_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% add_leE
tff(fact_2323_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))) ).

% le_add1
tff(fact_2324_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% le_add2
tff(fact_2325_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% add_leD1
tff(fact_2326_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% add_leD2
tff(fact_2327_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_2328_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_le_mono
tff(fact_2329_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_le_mono1
tff(fact_2330_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_le_add1
tff(fact_2331_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_le_add2
tff(fact_2332_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_2333_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% add_mult_distrib
tff(fact_2334_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% add_mult_distrib2
tff(fact_2335_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_2336_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_2337_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_2338_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_2339_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2340_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2341_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2342_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2343_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing
tff(fact_2344_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_increasing
tff(fact_2345_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing
tff(fact_2346_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing
tff(fact_2347_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2348_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2349_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2350_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2351_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing2
tff(fact_2352_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_increasing2
tff(fact_2353_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing2
tff(fact_2354_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing2
tff(fact_2355_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2356_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2357_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2358_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2359_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_2360_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_2361_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_nonpos
tff(fact_2362_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_2363_add__nonneg__eq__0__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y) = zero_zero_real )
        <=> ( ( X3 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2364_add__nonneg__eq__0__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y) = zero_zero_rat )
        <=> ( ( X3 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2365_add__nonneg__eq__0__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) = zero_zero_nat )
        <=> ( ( X3 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2366_add__nonneg__eq__0__iff,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y) = zero_zero_int )
        <=> ( ( X3 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2367_add__nonpos__eq__0__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y) = zero_zero_real )
        <=> ( ( X3 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2368_add__nonpos__eq__0__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y) = zero_zero_rat )
        <=> ( ( X3 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2369_add__nonpos__eq__0__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) = zero_zero_nat )
        <=> ( ( X3 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2370_add__nonpos__eq__0__iff,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y) = zero_zero_int )
        <=> ( ( X3 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2371_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_2372_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_2373_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_2374_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_2375_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_pos
tff(fact_2376_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_pos_pos
tff(fact_2377_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_pos
tff(fact_2378_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_pos
tff(fact_2379_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ ! [C4: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) )
           => ( C4 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_2380_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% pos_add_strict
tff(fact_2381_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% pos_add_strict
tff(fact_2382_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% pos_add_strict
tff(fact_2383_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% pos_add_strict
tff(fact_2384_add__less__zeroD,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_2385_add__less__zeroD,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y)),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_2386_add__less__zeroD,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_2387_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2388_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2389_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2390_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2391_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2392_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2393_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2394_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2395_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_le_less_mono
tff(fact_2396_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_2397_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_2398_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_le_less_mono
tff(fact_2399_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_less_le_mono
tff(fact_2400_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_2401_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_2402_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_less_le_mono
tff(fact_2403_less__add__one,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real))) ).

% less_add_one
tff(fact_2404_less__add__one,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat))) ).

% less_add_one
tff(fact_2405_less__add__one,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))) ).

% less_add_one
tff(fact_2406_less__add__one,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))) ).

% less_add_one
tff(fact_2407_add__mono1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2408_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat))) ) ).

% add_mono1
tff(fact_2409_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2410_add__mono1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_2411_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_2412_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_2413_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_2414_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_2415_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).

% less_add_Suc2
tff(fact_2416_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).

% less_add_Suc1
tff(fact_2417_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q3: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_2418_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_2419_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_2420_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_2421_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_2422_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M2: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M2)),aa_nat_nat(F,N2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_2423_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc
tff(fact_2424_mlex__snd__decrI,axiom,
    ! [A: nat,A5: nat,B: nat,B5: nat,N7: nat] :
      ( ( A = A5 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),B5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),N7)),B5))) ) ) ).

% mlex_snd_decrI
tff(fact_2425_mlex__fst__decrI,axiom,
    ! [A: nat,A5: nat,B: nat,N7: nat,B5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N7))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B5),N7))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),N7)),B5))) ) ) ) ).

% mlex_fst_decrI
tff(fact_2426_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N7: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N7))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7))) ) ) ).

% mlex_bound
tff(fact_2427_mlex__leI,axiom,
    ! [A: nat,A5: nat,B: nat,B5: nat,N7: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),B5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),N7)),B5))) ) ) ).

% mlex_leI
tff(fact_2428_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Uz),Va2,Vb),Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_2429_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_2430_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X3: fun_VE5569070535936029429omplex,Y: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_ep(I5),X3))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_ep(I5),Y))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_eq(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2431_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_er(I5),X3))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_er(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_es(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2432_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_et(I5),X3))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_et(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_eu(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2433_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_ev(I5),X3))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_ev(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_ew(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2434_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ex(I5),X3))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ex(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_ey(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2435_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X3: fun_Co1900716531650368264omplex,Y: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aTP_Lamm_ez(I5),X3))))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aTP_Lamm_ez(I5),Y))))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aa_fun6901856055265004766r_bool(aTP_Lamm_fa(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2436_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X3: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_fb(I5),X3))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_fb(I5),Y))))
       => pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_fc(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2437_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fd(I5),X3))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fd(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_fe(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2438_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ff(I5),X3))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ff(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_fg(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2439_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_fh(I5),X3))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_fh(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_fi(I5),X3),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_2440_field__le__epsilon,axiom,
    ! [X3: real,Y: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ).

% field_le_epsilon
tff(fact_2441_field__le__epsilon,axiom,
    ! [X3: rat,Y: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y)) ) ).

% field_le_epsilon
tff(fact_2442_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2443_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2444_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2445_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2446_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing
tff(fact_2447_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C))) ) ) ).

% add_strict_increasing
tff(fact_2448_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing
tff(fact_2449_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing
tff(fact_2450_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2451_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2452_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2453_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2454_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_2455_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_2456_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_2457_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_2458_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2459_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2460_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2461_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2462_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_2463_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_2464_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_2465_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_2466_sum__squares__ge__zero,axiom,
    ! [X3: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2467_sum__squares__ge__zero,axiom,
    ! [X3: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2468_sum__squares__ge__zero,axiom,
    ! [X3: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2469_zero__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ).

% zero_less_two
tff(fact_2470_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ).

% zero_less_two
tff(fact_2471_zero__less__two,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% zero_less_two
tff(fact_2472_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).

% zero_less_two
tff(fact_2473_not__sum__squares__lt__zero,axiom,
    ! [X3: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_2474_not__sum__squares__lt__zero,axiom,
    ! [X3: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_2475_not__sum__squares__lt__zero,axiom,
    ! [X3: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_2476_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X3))
     => ( ! [Uv2: bool] : X3 != aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2)
       => ( ! [Uu2: bool] : X3 != aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Uz2),Va3,Vb2),Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_2477_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X3))
     => ( ( X3 != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_2478_convex__bound__le,axiom,
    ! [X3: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2479_convex__bound__le,axiom,
    ! [X3: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2480_convex__bound__le,axiom,
    ! [X3: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2481_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X3))
      <=> pp(Y) )
     => ( ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Uz2),Va3,Vb2),Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_2482_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,Va2: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_bool_VEBT_VEBT(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),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(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_2483_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: bool,B: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_bool_VEBT_VEBT(vEBT_Leaf(Uu),B)),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_2484_convex__bound__lt,axiom,
    ! [X3: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2485_convex__bound__lt,axiom,
    ! [X3: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2486_convex__bound__lt,axiom,
    ! [X3: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2487_option_Osize__gen_I2_J,axiom,
    ! [X3: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(X3,aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X3,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_2488_option_Osize__gen_I2_J,axiom,
    ! [X3: fun_nat_nat,X2: nat] : size_option_nat(X3,aa_nat_option_nat(some_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(X3,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_2489_option_Osize__gen_I2_J,axiom,
    ! [X3: fun_num_nat,X2: num] : size_option_num(X3,aa_num_option_num(some_num,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X3,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_2490_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: bool,Uw: bool] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_bool_VEBT_VEBT(vEBT_Leaf(A),Uw)),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_2491_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,T),X3)),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_2492_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,T),X3)),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_2493_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,T),X3)),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_2494_sum__squares__eq__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2495_sum__squares__eq__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2496_sum__squares__eq__zero__iff,axiom,
    ! [X3: 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,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2497_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_2498_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_2499_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_2500_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X3: fun_Pr2477239776880934807at_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X3,Xa,Xb) = Y )
     => ( accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(X3,produc488173922507101015at_nat(Xa,Xb)))
       => ( ( ( Xa = none_P5556105721700978146at_nat )
           => ( ( Y = none_P5556105721700978146at_nat )
             => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(X3,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Xb))) ) )
         => ( ! [V2: product_prod_nat_nat] :
                ( ( Xa = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2) )
               => ( ( Xb = none_P5556105721700978146at_nat )
                 => ( ( Y = none_P5556105721700978146at_nat )
                   => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(X3,produc488173922507101015at_nat(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),none_P5556105721700978146at_nat))) ) ) )
           => ~ ! [A3: product_prod_nat_nat] :
                  ( ( Xa = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,A3) )
                 => ! [B2: product_prod_nat_nat] :
                      ( ( Xb = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B2) )
                     => ( ( Y = aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X3,A3),B2)) )
                       => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(X3,produc488173922507101015at_nat(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,A3),aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_2501_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X3: fun_num_fun_num_num,Xa: option_num,Xb: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X3,Xa,Xb) = Y )
     => ( accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(X3,produc8585076106096196333on_num(Xa,Xb)))
       => ( ( ( Xa = none_num )
           => ( ( Y = none_num )
             => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(X3,produc8585076106096196333on_num(none_num,Xb))) ) )
         => ( ! [V2: num] :
                ( ( Xa = aa_num_option_num(some_num,V2) )
               => ( ( Xb = none_num )
                 => ( ( Y = none_num )
                   => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(X3,produc8585076106096196333on_num(aa_num_option_num(some_num,V2),none_num))) ) ) )
           => ~ ! [A3: num] :
                  ( ( Xa = aa_num_option_num(some_num,A3) )
                 => ! [B2: num] :
                      ( ( Xb = aa_num_option_num(some_num,B2) )
                     => ( ( Y = aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(X3,A3),B2)) )
                       => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(X3,produc8585076106096196333on_num(aa_num_option_num(some_num,A3),aa_num_option_num(some_num,B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_2502_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X3: fun_nat_fun_nat_nat,Xa: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X3),Xa),Xb) = Y )
     => ( accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(X3,produc5098337634421038937on_nat(Xa,Xb)))
       => ( ( ( Xa = none_nat )
           => ( ( Y = none_nat )
             => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(X3,produc5098337634421038937on_nat(none_nat,Xb))) ) )
         => ( ! [V2: nat] :
                ( ( Xa = aa_nat_option_nat(some_nat,V2) )
               => ( ( Xb = none_nat )
                 => ( ( Y = none_nat )
                   => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(X3,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,V2),none_nat))) ) ) )
           => ~ ! [A3: nat] :
                  ( ( Xa = aa_nat_option_nat(some_nat,A3) )
                 => ! [B2: nat] :
                      ( ( Xb = aa_nat_option_nat(some_nat,B2) )
                     => ( ( Y = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(X3,A3),B2)) )
                       => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(X3,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,A3),aa_nat_option_nat(some_nat,B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_2503_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,T),X3)),one_one_nat)) ) ) ).

% minNull_delete_time_bound'
tff(fact_2504_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(T),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2505_discrete,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B)) ) ).

% discrete
tff(fact_2506_discrete,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B)) ) ).

% discrete
tff(fact_2507_sum__squares__gt__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))
    <=> ( ( X3 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2508_sum__squares__gt__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))))
    <=> ( ( X3 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2509_sum__squares__gt__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X3 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2510_sum__squares__le__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real))
    <=> ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2511_sum__squares__le__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat))
    <=> ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2512_sum__squares__le__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
    <=> ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2513_in__measure,axiom,
    ! [X3: nat,Y: nat,F: fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y),measure_nat(F))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y))) ) ).

% in_measure
tff(fact_2514_in__measure,axiom,
    ! [X3: int,Y: int,F: fun_int_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y),measure_int(F))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(F,Y))) ) ).

% in_measure
tff(fact_2515_in__measure,axiom,
    ! [X3: code_integer,Y: code_integer,F: fun_Code_integer_nat] :
      ( member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X3),Y),measure_Code_integer(F))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F,X3)),aa_Code_integer_nat(F,Y))) ) ).

% in_measure
tff(fact_2516_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,X3) = Y )
     => ( ! [A3: bool] :
            ( ? [B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A3),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_2517_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X3)),N)) ) ).

% delete_pres_valid
tff(fact_2518_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X3)),Y))
      <=> ( ( X3 != Y )
          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y)) ) ) ) ).

% dele_member_cont_corr
tff(fact_2519_vebt__inserti__refines,axiom,
    ! [Ti: vEBT_VEBTi,X3: nat,T: vEBT_VEBT] : refine5565527176597971370_VEBTi(vEBT_vebt_inserti(Ti,X3),vEBT_V3964819847710782039nserti(T,Ti,X3)) ).

% vebt_inserti_refines
tff(fact_2520_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_2521_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(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_2522_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,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_2523_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),X3))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A3),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_2524_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X3: nat] : hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_V3964819847710782039nserti(T,Ti,X3),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X3))) ).

% vebt_inserti'_rf_abstr
tff(fact_2525_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X3: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_2526_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X3: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst),Smry)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst),Smry) ).

% vebt_delete.simps(5)
tff(fact_2527_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_fj(A2),N))) ) ).

% finite_lists_length_le
tff(fact_2528_finite__lists__length__le,axiom,
    ! [A2: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_fk(A2),N))) ) ).

% finite_lists_length_le
tff(fact_2529_finite__lists__length__le,axiom,
    ! [A2: set_Code_integer,N: nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat3680331675976215733r_bool(aTP_Lamm_fl(A2),N))) ) ).

% finite_lists_length_le
tff(fact_2530_finite__lists__length__le,axiom,
    ! [A2: set_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => finite306553202115118035t_real(collect_list_real(aa_nat4570152432052498854l_bool(aTP_Lamm_fm(A2),N))) ) ).

% finite_lists_length_le
tff(fact_2531_finite__lists__length__le,axiom,
    ! [A2: set_bool,N: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fn(A2),N))) ) ).

% finite_lists_length_le
tff(fact_2532_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_fo(A2),N))) ) ).

% finite_lists_length_le
tff(fact_2533_finite__lists__length__le,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_fp(A2),N))) ) ).

% finite_lists_length_le
tff(fact_2534_vebt__delete_Osimps_I2_J,axiom,
    ! [A: bool,B: bool] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_VEBT_VEBT(vEBT_Leaf(A),fFalse) ).

% vebt_delete.simps(2)
tff(fact_2535_divides__aux__eq,axiom,
    ! [Q4: code_integer,R2: code_integer] :
      ( unique5706413561485394159nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Q4),R2))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_2536_divides__aux__eq,axiom,
    ! [Q4: nat,R2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2)))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_2537_divides__aux__eq,axiom,
    ! [Q4: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_2538_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_fq(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_2539_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_fr(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_2540_finite__lists__length__eq,axiom,
    ! [A2: set_Code_integer,N: nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat3680331675976215733r_bool(aTP_Lamm_fs(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_2541_finite__lists__length__eq,axiom,
    ! [A2: set_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => finite306553202115118035t_real(collect_list_real(aa_nat4570152432052498854l_bool(aTP_Lamm_ft(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_2542_finite__lists__length__eq,axiom,
    ! [A2: set_bool,N: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fu(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_2543_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_fv(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_2544_finite__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_fw(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_2545_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_2546_add__shift,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_nat_option_nat(some_nat,X3)),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,Z) ) ) ).

% add_shift
tff(fact_2547_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_real] : aa_list_real_nat(size_size_list_real,Xs2) = N ).

% Ex_list_of_length
tff(fact_2548_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_bool] : aa_list_bool_nat(size_size_list_bool,Xs2) = N ).

% Ex_list_of_length
tff(fact_2549_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : aa_list_nat_nat(size_size_list_nat,Xs2) = N ).

% Ex_list_of_length
tff(fact_2550_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : aa_list_int_nat(size_size_list_int,Xs2) = N ).

% Ex_list_of_length
tff(fact_2551_neq__if__length__neq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) != aa_list_real_nat(size_size_list_real,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_2552_neq__if__length__neq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) != aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_2553_neq__if__length__neq,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) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_2554_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) != aa_list_int_nat(size_size_list_int,Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_2555_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( finite306553202115118035t_real(M8)
     => ? [N2: nat] :
        ! [X5: list_real] :
          ( member_list_real(X5,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,X5)),N2)) ) ) ).

% finite_maxlen
tff(fact_2556_finite__maxlen,axiom,
    ! [M8: set_list_bool] :
      ( finite7866827932528546246t_bool(M8)
     => ? [N2: nat] :
        ! [X5: list_bool] :
          ( member_list_bool(X5,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_bool_nat(size_size_list_bool,X5)),N2)) ) ) ).

% finite_maxlen
tff(fact_2557_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( finite8100373058378681591st_nat(M8)
     => ? [N2: nat] :
        ! [X5: list_nat] :
          ( pp(aa_set_list_nat_bool(member_list_nat(X5),M8))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,X5)),N2)) ) ) ).

% finite_maxlen
tff(fact_2558_finite__maxlen,axiom,
    ! [M8: set_list_int] :
      ( finite3922522038869484883st_int(M8)
     => ? [N2: nat] :
        ! [X5: list_int] :
          ( member_list_int(X5,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_int_nat(size_size_list_int,X5)),N2)) ) ) ).

% finite_maxlen
tff(fact_2559_length__induct,axiom,
    ! [P: fun_list_real_bool,Xs: list_real] :
      ( ! [Xs2: list_real] :
          ( ! [Ys2: list_real] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,Ys2)),aa_list_real_nat(size_size_list_real,Xs2)))
             => pp(aa_list_real_bool(P,Ys2)) )
         => pp(aa_list_real_bool(P,Xs2)) )
     => pp(aa_list_real_bool(P,Xs)) ) ).

% length_induct
tff(fact_2560_length__induct,axiom,
    ! [P: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs2: list_bool] :
          ( ! [Ys2: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_bool_nat(size_size_list_bool,Ys2)),aa_list_bool_nat(size_size_list_bool,Xs2)))
             => pp(aa_list_bool_bool(P,Ys2)) )
         => pp(aa_list_bool_bool(P,Xs2)) )
     => pp(aa_list_bool_bool(P,Xs)) ) ).

% length_induct
tff(fact_2561_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ys2)),aa_list_nat_nat(size_size_list_nat,Xs2)))
             => pp(aa_list_nat_bool(P,Ys2)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_2562_length__induct,axiom,
    ! [P: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_int_nat(size_size_list_int,Ys2)),aa_list_int_nat(size_size_list_int,Xs2)))
             => pp(aa_list_int_bool(P,Ys2)) )
         => pp(aa_list_int_bool(P,Xs2)) )
     => pp(aa_list_int_bool(P,Xs)) ) ).

% length_induct
tff(fact_2563_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B3))
    <=> ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),set_set_nat2(Xs)))
         => pp(aa_set_set_nat_bool(member_set_nat(X4),B3)) ) ) ).

% subset_code(1)
tff(fact_2564_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),B3))
    <=> ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),B3)) ) ) ).

% subset_code(1)
tff(fact_2565_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Xs)),B3))
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_set_nat_bool(member_nat(X4),B3)) ) ) ).

% subset_code(1)
tff(fact_2566_subset__code_I1_J,axiom,
    ! [Xs: list_real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),B3))
    <=> ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_set_real_bool(member_real(X4),B3)) ) ) ).

% subset_code(1)
tff(fact_2567_subset__code_I1_J,axiom,
    ! [Xs: list_bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs)),B3))
    <=> ! [X4: bool] :
          ( pp(aa_set_bool_bool(member_bool(X4),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_set_bool_bool(member_bool(X4),B3)) ) ) ).

% subset_code(1)
tff(fact_2568_subset__code_I1_J,axiom,
    ! [Xs: list_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Xs)),B3))
    <=> ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),aa_list_int_set_int(set_int2,Xs)))
         => pp(aa_set_int_bool(member_int(X4),B3)) ) ) ).

% subset_code(1)
tff(fact_2569_length__pos__if__in__set,axiom,
    ! [X3: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2570_length__pos__if__in__set,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2571_length__pos__if__in__set,axiom,
    ! [X3: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs))) ) ).

% length_pos_if_in_set
tff(fact_2572_length__pos__if__in__set,axiom,
    ! [X3: bool,Xs: list_bool] :
      ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_bool_nat(size_size_list_bool,Xs))) ) ).

% length_pos_if_in_set
tff(fact_2573_length__pos__if__in__set,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ) ).

% length_pos_if_in_set
tff(fact_2574_length__pos__if__in__set,axiom,
    ! [X3: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_int_nat(size_size_list_int,Xs))) ) ).

% length_pos_if_in_set
tff(fact_2575_vebt__delete_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,N: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_bool_VEBT_VEBT(vEBT_Leaf(A),B) ).

% vebt_delete.simps(3)
tff(fact_2576_vebt__delete_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),zero_zero_nat) = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),B) ).

% vebt_delete.simps(1)
tff(fact_2577_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) ).

% vebt_delete.simps(4)
tff(fact_2578_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary) ).

% vebt_insert.simps(4)
tff(fact_2579_vebt__insert_Osimps_I1_J,axiom,
    ! [X3: nat,A: bool,B: bool] :
      ( ( ( X3 = zero_zero_nat )
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),X3) = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),B) ) )
      & ( ( X3 != zero_zero_nat )
       => ( ( ( X3 = one_one_nat )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),X3) = aa_bool_VEBT_VEBT(vEBT_Leaf(A),fTrue) ) )
          & ( ( X3 != one_one_nat )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),X3) = aa_bool_VEBT_VEBT(vEBT_Leaf(A),B) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_2580_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) ).

% vebt_insert.simps(3)
tff(fact_2581_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2) ).

% vebt_insert.simps(2)
tff(fact_2582_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
             => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) )
          & ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_2583_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X3))
      <=> pp(Y) )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X3))
       => ( ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
           => ( pp(Y)
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse))) ) )
         => ( ! [Uv2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2) )
               => ( ~ pp(Y)
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2))) ) )
           => ( ! [Uu2: bool] :
                  ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue) )
                 => ( ~ pp(Y)
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue))) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( pp(Y)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Uz2),Va3,Vb2),Vc2) )
                     => ( ~ pp(Y)
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Uz2),Va3,Vb2),Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_2584_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2585_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2586_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2587_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2588_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),C)) != aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2589_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,X3) = Y )
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_2590_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_nat_nat,A1: nat,A22: nat,A32: nat,P: fun_fu7164412255009898145t_bool] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(A0,produc487386426758144856at_nat(A1,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A22),A32))))
     => ( ! [F2: fun_nat_fun_nat_nat,A3: nat,B2: nat,Acc: nat] :
            ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Acc))))
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aa_fun2305595801590362634t_bool(P,F2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),one_one_nat)),B2),aa_nat_nat(aa_nat_fun_nat_nat(F2,A3),Acc))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aa_fun2305595801590362634t_bool(P,F2),A3),B2),Acc)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aa_fun2305595801590362634t_bool(P,A0),A1),A22),A32)) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_2591_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_num_num,A1: nat,A22: nat,A32: num,P: fun_fu2950424802767353155m_bool] :
      ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(A0,produc1195630363706982562at_num(A1,aa_num5026351239867140595at_num(product_Pair_nat_num(A22),A32))))
     => ( ! [F2: fun_nat_fun_num_num,A3: nat,B2: nat,Acc: num] :
            ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(F2,produc1195630363706982562at_num(A3,aa_num5026351239867140595at_num(product_Pair_nat_num(B2),Acc))))
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A3))
               => pp(aa_num_bool(aa_nat_fun_num_bool(aa_nat2608338788294301521m_bool(aa_fun7437090807486079916m_bool(P,F2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),one_one_nat)),B2),aa_num_num(aa_nat_fun_num_num(F2,A3),Acc))) )
             => pp(aa_num_bool(aa_nat_fun_num_bool(aa_nat2608338788294301521m_bool(aa_fun7437090807486079916m_bool(P,F2),A3),B2),Acc)) ) )
       => pp(aa_num_bool(aa_nat_fun_num_bool(aa_nat2608338788294301521m_bool(aa_fun7437090807486079916m_bool(P,A0),A1),A22),A32)) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_2592_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_13)) ) ).

% not_min_Null_member
tff(fact_2593_maxbmo,axiom,
    ! [T: vEBT_VEBT,X3: nat] :
      ( ( aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,T) = aa_nat_option_nat(some_nat,X3) )
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X3)) ) ).

% maxbmo
tff(fact_2594_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X3)),Y))
      <=> ( ( X3 != Y )
          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),Y)) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_2595_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X3))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X3)) ) ) ).

% both_member_options_equiv_member
tff(fact_2596_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X3))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X3)) ) ) ).

% valid_member_both_member_options
tff(fact_2597_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X3))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,T),X3))
        | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,T),X3)) ) ) ).

% both_member_options_def
tff(fact_2598_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : aa_VEBT_VEBT_set_nat(vEBT_set_vebt,T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_2599_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(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_2600_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_2601_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,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_2602_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_2603_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_2604_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_2605_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_2606_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_2607_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2608_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2609_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2610_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2611_crossproduct__noteq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),D)) != aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),D)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2612_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X3: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y)) )
    <=> ( ( W2 = X3 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2613_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X3: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y)) )
    <=> ( ( W2 = X3 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2614_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X3: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y)) )
    <=> ( ( W2 = X3 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2615_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X3: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X3),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y)) )
    <=> ( ( W2 = X3 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2616_crossproduct__eq,axiom,
    ! [W2: complex,Y: complex,X3: complex,Z: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W2),Y)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W2),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),Y)) )
    <=> ( ( W2 = X3 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2617_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X3))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X3))
       => ( ! [Uv2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2) )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),Uv2))) )
         => ( ! [Uu2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),fTrue))) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Uz2),Va3,Vb2),Vc2) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,Uz2),Va3,Vb2),Vc2))) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_2618_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X3))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X3))
       => ( ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
           => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse))) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2))) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_2619_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),X3))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2),Uz2))) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_2620_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X3: fun_nat_fun_num_num,Xa: nat,Xb: nat,Xc: num,Y: num] :
      ( ( aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(X3,Xa),Xb),Xc) = Y )
     => ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(X3,produc1195630363706982562at_num(Xa,aa_num5026351239867140595at_num(product_Pair_nat_num(Xb),Xc))))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
               => ( Y = Xc ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
               => ( Y = aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(X3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_num_num(aa_nat_fun_num_num(X3,Xa),Xc)) ) ) )
           => ~ accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(X3,produc1195630363706982562at_num(Xa,aa_num5026351239867140595at_num(product_Pair_nat_num(Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_2621_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X3: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X3,Xa),Xb),Xc) = Y )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X3,produc487386426758144856at_nat(Xa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc))))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
               => ( Y = Xc ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X3,Xa),Xc)) ) ) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X3,produc487386426758144856at_nat(Xa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_2622_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_num_num,A: nat,B: nat,Acc2: num] :
      ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(F,produc1195630363706982562at_num(A,aa_num5026351239867140595at_num(product_Pair_nat_num(B),Acc2))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ( aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(F,A),B),Acc2) = Acc2 ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ( aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(F,A),B),Acc2) = aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_num_num(aa_nat_fun_num_num(F,A),Acc2)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_2623_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F,produc487386426758144856at_nat(A,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Acc2))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = Acc2 ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_2624_foldr__same__int,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ! [X: nat,Y3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => ( pp(aa_set_nat_bool(member_nat(Y3),aa_list_nat_set_nat(set_nat2,Xs)))
           => ( X = Y3 ) ) )
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(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_2625_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_2626_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,T),X3)),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_2627_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool,N: nat] :
      ( ! [X: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
         => pp(aa_VEBT_VEBTi_bool(P,X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
       => pp(aa_VEBT_VEBTi_bool(P,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N))) ) ) ).

% inthall
tff(fact_2628_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,N: nat] :
      ( ! [X: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
         => pp(aa_set_nat_bool(P,X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),N))) ) ) ).

% inthall
tff(fact_2629_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N))) ) ) ).

% inthall
tff(fact_2630_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N: nat] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_real_bool(P,X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% inthall
tff(fact_2631_inthall,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,N: nat] :
      ( ! [X: bool] :
          ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_bool_bool(P,X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% inthall
tff(fact_2632_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_nat_bool(P,X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_2633_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N: nat] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
         => pp(aa_int_bool(P,X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% inthall
tff(fact_2634_set__n__lists,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : set_list_VEBT_VEBT2(n_lists_VEBT_VEBT(N,Xs)) = collec5608196760682091941T_VEBT(aa_lis561826058711847838T_bool(aTP_Lamm_fx(N),Xs)) ).

% set_n_lists
tff(fact_2635_set__n__lists,axiom,
    ! [N: nat,Xs: list_real] : set_list_real2(n_lists_real(N,Xs)) = collect_list_real(aa_lis8290120702490534970l_bool(aTP_Lamm_fy(N),Xs)) ).

% set_n_lists
tff(fact_2636_set__n__lists,axiom,
    ! [N: nat,Xs: list_bool] : set_list_bool2(n_lists_bool(N,Xs)) = collect_list_bool(aa_lis3148754934519358302l_bool(aTP_Lamm_fz(N),Xs)) ).

% set_n_lists
tff(fact_2637_set__n__lists,axiom,
    ! [N: nat,Xs: list_nat] : set_list_nat2(n_lists_nat(N,Xs)) = collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_ga(N),Xs)) ).

% set_n_lists
tff(fact_2638_set__n__lists,axiom,
    ! [N: nat,Xs: list_int] : set_list_int2(n_lists_int(N,Xs)) = collect_list_int(aa_lis3210702172589064762t_bool(aTP_Lamm_gb(N),Xs)) ).

% set_n_lists
tff(fact_2639_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B2))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B2),A3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),zero_zero_nat))
       => ( ! [A3: nat,B2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_2640_foldr__one,axiom,
    ! [D: nat,Ys: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D))) ).

% foldr_one
tff(fact_2641_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Ys),I2))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),C)),aa_list_nat_nat(size_size_list_nat,Ys))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D))) ) ) ) ).

% foldr_mono
tff(fact_2642_foldr__length,axiom,
    ! [L: list_real] : aa_nat_nat(foldr_real_nat(aTP_Lamm_gc,L),zero_zero_nat) = aa_list_real_nat(size_size_list_real,L) ).

% foldr_length
tff(fact_2643_foldr__length,axiom,
    ! [L: list_bool] : aa_nat_nat(foldr_bool_nat(aTP_Lamm_gd,L),zero_zero_nat) = aa_list_bool_nat(size_size_list_bool,L) ).

% foldr_length
tff(fact_2644_foldr__length,axiom,
    ! [L: list_nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_ge,L),zero_zero_nat) = aa_list_nat_nat(size_size_list_nat,L) ).

% foldr_length
tff(fact_2645_foldr__length,axiom,
    ! [L: list_int] : aa_nat_nat(foldr_int_nat(aTP_Lamm_gf,L),zero_zero_nat) = aa_list_int_nat(size_size_list_int,L) ).

% foldr_length
tff(fact_2646_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] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2647_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] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2648_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_real_nat(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_2649_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_bool_nat(size_size_list_bool,Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I3))
            <=> pp(aa_nat_bool(nth_bool(Ys),I3)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2650_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] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2651_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(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_2652_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na307722919519257988i_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_12: vEBT_VEBTi] : pp(aa_VEBT_VEBTi_bool(aa_nat6176657665065580909i_bool(P,I3),X_12)) )
    <=> ? [Xs3: list_VEBT_VEBTi] :
          ( ( size_s7982070591426661849_VEBTi(Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_VEBT_VEBTi_bool(aa_nat6176657665065580909i_bool(P,I3),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_2653_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na2351039247863542899T_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_12: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I3),X_12)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I3),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_2654_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6801867027684862029l_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_12: real] : pp(aa_real_bool(aa_nat_fun_real_bool(P,I3),X_12)) )
    <=> ? [Xs3: list_real] :
          ( ( aa_list_real_nat(size_size_list_real,Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_real_bool(aa_nat_fun_real_bool(P,I3),aa_nat_real(nth_real(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_2655_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na9137082442855928016l_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_12: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I3),X_12)) )
    <=> ? [Xs3: list_bool] :
          ( ( aa_list_bool_nat(size_size_list_bool,Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I3),aa_nat_bool(nth_bool(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_2656_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),X_12)) )
    <=> ? [Xs3: list_nat] :
          ( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(nth_nat(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_2657_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_12: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I3),X_12)) )
    <=> ? [Xs3: list_int] :
          ( ( aa_list_int_nat(size_size_list_int,Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P,I3),aa_nat_int(nth_int(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_2658_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2659_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2660_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_real_nat(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_2661_nth__equalityI,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_bool_nat(size_size_list_bool,Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I2))
            <=> pp(aa_nat_bool(nth_bool(Ys),I2)) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_2662_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] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2663_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_int_nat(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_2664_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_VE3025410955681380100t_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: vEBT_VEBTi] : pp(aa_nat_bool(aa_VEB8502397618584205677t_bool(P,Li),I2)) )
     => ~ ! [L2: list_VEBT_VEBTi] :
            ( ( size_s7982070591426661849_VEBTi(L2) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_VEB8502397618584205677t_bool(P,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L2),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_2665_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_VE3763956218935105199t_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: vEBT_VEBT] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(P,Li),I2)) )
     => ~ ! [L2: list_VEBT_VEBT] :
            ( ( size_s6755466524823107622T_VEBT(L2) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L2),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_2666_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_re9121486324101951821t_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: real] : pp(aa_nat_bool(aa_real_fun_nat_bool(P,Li),I2)) )
     => ~ ! [L2: list_real] :
            ( ( aa_list_real_nat(size_size_list_real,L2) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_real_fun_nat_bool(P,aa_nat_real(nth_real(L2),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_2667_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_bo3288154511880838028t_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: bool] : pp(aa_nat_bool(aa_bool_fun_nat_bool(P,Li),I2)) )
     => ~ ! [L2: list_bool] :
            ( ( aa_list_bool_nat(size_size_list_bool,L2) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_bool_fun_nat_bool(P,aa_nat_bool(nth_bool(L2),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_2668_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,Li),I2)) )
     => ~ ! [L2: list_nat] :
            ( ( aa_list_nat_nat(size_size_list_nat,L2) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(nth_nat(L2),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_2669_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_int_fun_nat_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => ? [Li: int] : pp(aa_nat_bool(aa_int_fun_nat_bool(P,Li),I2)) )
     => ~ ! [L2: list_int] :
            ( ( aa_list_int_nat(size_size_list_int,L2) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_int_fun_nat_bool(P,aa_nat_int(nth_int(L2),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_2670_length__n__lists__elem,axiom,
    ! [Ys: list_real,N: nat,Xs: list_real] :
      ( member_list_real(Ys,set_list_real2(n_lists_real(N,Xs)))
     => ( aa_list_real_nat(size_size_list_real,Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_2671_length__n__lists__elem,axiom,
    ! [Ys: list_bool,N: nat,Xs: list_bool] :
      ( member_list_bool(Ys,set_list_bool2(n_lists_bool(N,Xs)))
     => ( aa_list_bool_nat(size_size_list_bool,Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_2672_length__n__lists__elem,axiom,
    ! [Ys: list_nat,N: nat,Xs: list_nat] :
      ( pp(aa_set_list_nat_bool(member_list_nat(Ys),set_list_nat2(n_lists_nat(N,Xs))))
     => ( aa_list_nat_nat(size_size_list_nat,Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_2673_length__n__lists__elem,axiom,
    ! [Ys: list_int,N: nat,Xs: list_int] :
      ( member_list_int(Ys,set_list_int2(n_lists_int(N,Xs)))
     => ( aa_list_int_nat(size_size_list_int,Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_2674_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => member_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N),set_VEBT_VEBTi2(Xs)) ) ).

% nth_mem
tff(fact_2675_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),N)),set_set_nat2(Xs))) ) ).

% nth_mem
tff(fact_2676_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))) ) ).

% nth_mem
tff(fact_2677_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
     => pp(aa_set_real_bool(member_real(aa_nat_real(nth_real(Xs),N)),aa_lis3769085826154824284t_real(set_real2,Xs))) ) ).

% nth_mem
tff(fact_2678_nth__mem,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
     => pp(aa_set_bool_bool(member_bool(aa_nat_bool(nth_bool(Xs),N)),aa_lis3587183946281582166t_bool(set_bool2,Xs))) ) ).

% nth_mem
tff(fact_2679_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),aa_list_nat_set_nat(set_nat2,Xs))) ) ).

% nth_mem
tff(fact_2680_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
     => pp(aa_set_int_bool(member_int(aa_nat_int(nth_int(Xs),N)),aa_list_int_set_int(set_int2,Xs))) ) ).

% nth_mem
tff(fact_2681_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( ! [X: vEBT_VEBTi] :
            ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
           => pp(aa_VEBT_VEBTi_bool(P,X)) )
       => pp(aa_VEBT_VEBTi_bool(P,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2682_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
           => pp(aa_VEBT_VEBT_bool(P,X)) )
       => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2683_list__ball__nth,axiom,
    ! [N: nat,Xs: list_real,P: fun_real_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
           => pp(aa_real_bool(P,X)) )
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2684_list__ball__nth,axiom,
    ! [N: nat,Xs: list_bool,P: fun_bool_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( ! [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
           => pp(aa_bool_bool(P,X)) )
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2685_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
           => pp(aa_nat_bool(P,X)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2686_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
           => pp(aa_int_bool(P,X)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_2687_in__set__conv__nth,axiom,
    ! [X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
          & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I3) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_2688_in__set__conv__nth,axiom,
    ! [X3: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs)))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I3) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_2689_in__set__conv__nth,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
          & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_2690_in__set__conv__nth,axiom,
    ! [X3: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_real_nat(size_size_list_real,Xs)))
          & ( aa_nat_real(nth_real(Xs),I3) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_2691_in__set__conv__nth,axiom,
    ! [X3: bool,Xs: list_bool] :
      ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_bool_nat(size_size_list_bool,Xs)))
          & ( pp(aa_nat_bool(nth_bool(Xs),I3))
          <=> pp(X3) ) ) ) ).

% in_set_conv_nth
tff(fact_2692_in__set__conv__nth,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I3) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_2693_in__set__conv__nth,axiom,
    ! [X3: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs)))
          & ( aa_nat_int(nth_int(Xs),I3) = X3 ) ) ) ).

% in_set_conv_nth
tff(fact_2694_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool,X3: vEBT_VEBTi] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
         => pp(aa_VEBT_VEBTi_bool(P,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2))) )
     => ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))
       => pp(aa_VEBT_VEBTi_bool(P,X3)) ) ) ).

% all_nth_imp_all_set
tff(fact_2695_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,X3: set_nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs)))
         => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),I2))) )
     => ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
       => pp(aa_set_nat_bool(P,X3)) ) ) ).

% all_nth_imp_all_set
tff(fact_2696_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X3: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2))) )
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
       => pp(aa_VEBT_VEBT_bool(P,X3)) ) ) ).

% all_nth_imp_all_set
tff(fact_2697_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X3: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_real_nat(size_size_list_real,Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I2))) )
     => ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
       => pp(aa_real_bool(P,X3)) ) ) ).

% all_nth_imp_all_set
tff(fact_2698_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,X3: bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_bool_nat(size_size_list_bool,Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I2))) )
     => ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
       => pp(aa_bool_bool(P,X3)) ) ) ).

% all_nth_imp_all_set
tff(fact_2699_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X3: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I2))) )
     => ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
       => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nth_imp_all_set
tff(fact_2700_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X3: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_int_nat(size_size_list_int,Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I2))) )
     => ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
       => pp(aa_int_bool(P,X3)) ) ) ).

% all_nth_imp_all_set
tff(fact_2701_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool] :
      ( ! [X4: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X4,set_VEBT_VEBTi2(Xs))
         => pp(aa_VEBT_VEBTi_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
         => pp(aa_VEBT_VEBTi_bool(P,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_2702_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_2703_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_real_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_real_nat(size_size_list_real,Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_2704_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P: fun_bool_bool] :
      ( ! [X4: bool] :
          ( pp(aa_set_bool_bool(member_bool(X4),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_bool_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_bool_nat(size_size_list_bool,Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_2705_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_nat_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_2706_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),aa_list_int_set_int(set_int2,Xs)))
         => pp(aa_int_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_2707_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool] :
      ( ! [X4: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X4,set_VEBT_VEBTi2(L))
         => pp(aa_VEBT_VEBTi_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(L)))
         => pp(aa_VEBT_VEBTi_bool(P,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I3))) ) ) ).

% all_set_conv_nth
tff(fact_2708_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L)))
         => pp(aa_VEBT_VEBT_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(L)))
         => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I3))) ) ) ).

% all_set_conv_nth
tff(fact_2709_all__set__conv__nth,axiom,
    ! [L: list_real,P: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),aa_lis3769085826154824284t_real(set_real2,L)))
         => pp(aa_real_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_real_nat(size_size_list_real,L)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(L),I3))) ) ) ).

% all_set_conv_nth
tff(fact_2710_all__set__conv__nth,axiom,
    ! [L: list_bool,P: fun_bool_bool] :
      ( ! [X4: bool] :
          ( pp(aa_set_bool_bool(member_bool(X4),aa_lis3587183946281582166t_bool(set_bool2,L)))
         => pp(aa_bool_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_bool_nat(size_size_list_bool,L)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(L),I3))) ) ) ).

% all_set_conv_nth
tff(fact_2711_all__set__conv__nth,axiom,
    ! [L: list_nat,P: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_list_nat_set_nat(set_nat2,L)))
         => pp(aa_nat_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,L)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(L),I3))) ) ) ).

% all_set_conv_nth
tff(fact_2712_all__set__conv__nth,axiom,
    ! [L: list_int,P: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),aa_list_int_set_int(set_int2,L)))
         => pp(aa_int_bool(P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,L)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(L),I3))) ) ) ).

% all_set_conv_nth
tff(fact_2713_size__list__estimation,axiom,
    ! [X3: int,Xs: list_int,Y: nat,F: fun_int_nat] :
      ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_int_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),size_list_int(F,Xs))) ) ) ).

% size_list_estimation
tff(fact_2714_size__list__estimation,axiom,
    ! [X3: set_nat,Xs: list_set_nat,Y: nat,F: fun_set_nat_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_set_nat_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),size_list_set_nat(F,Xs))) ) ) ).

% size_list_estimation
tff(fact_2715_size__list__estimation,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_list_nat_nat(size_list_nat(F),Xs))) ) ) ).

% size_list_estimation
tff(fact_2716_size__list__estimation,axiom,
    ! [X3: real,Xs: list_real,Y: nat,F: fun_real_nat] :
      ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_real_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),size_list_real(F,Xs))) ) ) ).

% size_list_estimation
tff(fact_2717_size__list__estimation,axiom,
    ! [X3: bool,Xs: list_bool,Y: nat,F: fun_bool_nat] :
      ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_bool_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),size_list_bool(F,Xs))) ) ) ).

% size_list_estimation
tff(fact_2718_size__list__estimation,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: nat,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_VEBT_VEBT_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),size_list_VEBT_VEBT(F,Xs))) ) ) ).

% size_list_estimation
tff(fact_2719_size__list__pointwise,axiom,
    ! [Xs: list_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X)),aa_int_nat(G,X))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_list_int(F,Xs)),size_list_int(G,Xs))) ) ).

% size_list_pointwise
tff(fact_2720_size__list__pointwise,axiom,
    ! [Xs: list_set_nat,F: fun_set_nat_nat,G: fun_set_nat_nat] :
      ( ! [X: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(F,X)),aa_set_nat_nat(G,X))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_list_set_nat(F,Xs)),size_list_set_nat(G,Xs))) ) ).

% size_list_pointwise
tff(fact_2721_size__list__pointwise,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X)),aa_nat_nat(G,X))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_list_nat(F),Xs)),aa_list_nat_nat(size_list_nat(G),Xs))) ) ).

% size_list_pointwise
tff(fact_2722_size__list__pointwise,axiom,
    ! [Xs: list_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X)),aa_real_nat(G,X))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_list_real(F,Xs)),size_list_real(G,Xs))) ) ).

% size_list_pointwise
tff(fact_2723_size__list__pointwise,axiom,
    ! [Xs: list_bool,F: fun_bool_nat,G: fun_bool_nat] :
      ( ! [X: bool] :
          ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(F,X)),aa_bool_nat(G,X))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_list_bool(F,Xs)),size_list_bool(G,Xs))) ) ).

% size_list_pointwise
tff(fact_2724_size__list__pointwise,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X)),aa_VEBT_VEBT_nat(G,X))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_list_VEBT_VEBT(F,Xs)),size_list_VEBT_VEBT(G,Xs))) ) ).

% size_list_pointwise
tff(fact_2725_size__list__estimation_H,axiom,
    ! [X3: int,Xs: list_int,Y: nat,F: fun_int_nat] :
      ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_int_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),size_list_int(F,Xs))) ) ) ).

% size_list_estimation'
tff(fact_2726_size__list__estimation_H,axiom,
    ! [X3: set_nat,Xs: list_set_nat,Y: nat,F: fun_set_nat_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_set_nat_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),size_list_set_nat(F,Xs))) ) ) ).

% size_list_estimation'
tff(fact_2727_size__list__estimation_H,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_list_nat_nat(size_list_nat(F),Xs))) ) ) ).

% size_list_estimation'
tff(fact_2728_size__list__estimation_H,axiom,
    ! [X3: real,Xs: list_real,Y: nat,F: fun_real_nat] :
      ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_real_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),size_list_real(F,Xs))) ) ) ).

% size_list_estimation'
tff(fact_2729_size__list__estimation_H,axiom,
    ! [X3: bool,Xs: list_bool,Y: nat,F: fun_bool_nat] :
      ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_bool_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),size_list_bool(F,Xs))) ) ) ).

% size_list_estimation'
tff(fact_2730_size__list__estimation_H,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: nat,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_VEBT_VEBT_nat(F,X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),size_list_VEBT_VEBT(F,Xs))) ) ) ).

% size_list_estimation'
tff(fact_2731_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: bool,B: bool,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),X3) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_2732_foldr__length__aux,axiom,
    ! [L: list_real,A: nat] : aa_nat_nat(foldr_real_nat(aTP_Lamm_gc,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_real_nat(size_size_list_real,L)) ).

% foldr_length_aux
tff(fact_2733_foldr__length__aux,axiom,
    ! [L: list_bool,A: nat] : aa_nat_nat(foldr_bool_nat(aTP_Lamm_gd,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_bool_nat(size_size_list_bool,L)) ).

% foldr_length_aux
tff(fact_2734_foldr__length__aux,axiom,
    ! [L: list_nat,A: nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_ge,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_2735_foldr__length__aux,axiom,
    ! [L: list_int,A: nat] : aa_nat_nat(foldr_int_nat(aTP_Lamm_gf,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_int_nat(size_size_list_int,L)) ).

% foldr_length_aux
tff(fact_2736_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,S2: vEBT_VEBT,X3: 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),S2)),X3) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_2737_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X3: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X3,Xa),Xb),Xc) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
         => ( Y = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb),Xa))
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X3,Xa),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2738_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B: nat,A: nat,F: fun_nat_fun_nat_nat,Acc2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = Acc2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2739_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,S2: vEBT_VEBT,X3: 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),S2)),X3) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_2740_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: 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,V)),TreeList),Summary)),X3) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_2741_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_13))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2742_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2743_vebt__maxtilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts2)))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(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_gg(I),Ts2),Tsi)) ) ).

% vebt_maxtilist
tff(fact_2744_vebt__mintilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts2)))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(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_gh(I),Ts2),Tsi)) ) ).

% vebt_mintilist
tff(fact_2745_foldr__zero,axiom,
    ! [Xs: list_nat,D: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs),I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2746_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_2747_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Bound: nat,I: nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_2748_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_real,F: fun_real_nat,Bound: nat,I: nat] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_real_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_real_nat(size_size_list_real,Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_2749_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_bool,F: fun_bool_nat,Bound: nat,I: nat] :
      ( ! [X: bool] :
          ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(F,X)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_bool_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_bool_nat(size_size_list_bool,Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_2750_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,Bound: nat,I: nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_2751_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_int,F: fun_int_nat,Bound: nat,I: nat] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_int_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_int_nat(size_size_list_int,Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_2752_length__mul__elem,axiom,
    ! [Xs: list_list_real,N: nat] :
      ( ! [X: list_real] :
          ( member_list_real(X,set_list_real2(Xs))
         => ( aa_list_real_nat(size_size_list_real,X) = N ) )
     => ( aa_list_real_nat(size_size_list_real,concat_real(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6660260683639930848t_real(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_2753_length__mul__elem,axiom,
    ! [Xs: list_list_bool,N: nat] :
      ( ! [X: list_bool] :
          ( member_list_bool(X,set_list_bool2(Xs))
         => ( aa_list_bool_nat(size_size_list_bool,X) = N ) )
     => ( aa_list_bool_nat(size_size_list_bool,concat_bool(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s9193230429364170767t_bool(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_2754_length__mul__elem,axiom,
    ! [Xs: list_list_nat,N: nat] :
      ( ! [X: list_nat] :
          ( pp(aa_set_list_nat_bool(member_list_nat(X),set_list_nat2(Xs)))
         => ( aa_list_nat_nat(size_size_list_nat,X) = N ) )
     => ( 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)),N) ) ) ).

% length_mul_elem
tff(fact_2755_length__mul__elem,axiom,
    ! [Xs: list_list_int,N: nat] :
      ( ! [X: list_int] :
          ( member_list_int(X,set_list_int2(Xs))
         => ( aa_list_int_nat(size_size_list_int,X) = N ) )
     => ( aa_list_int_nat(size_size_list_int,concat_int(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s533118279054570080st_int(Xs)),N) ) ) ).

% length_mul_elem
tff(fact_2756_ran__nth__set__encoding__conv,axiom,
    ! [L: list_VEBT_VEBTi] : ran_nat_VEBT_VEBTi(aTP_Lamm_gi(L)) = set_VEBT_VEBTi2(L) ).

% ran_nth_set_encoding_conv
tff(fact_2757_ran__nth__set__encoding__conv,axiom,
    ! [L: list_VEBT_VEBT] : ran_nat_VEBT_VEBT(aTP_Lamm_gj(L)) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L) ).

% ran_nth_set_encoding_conv
tff(fact_2758_ran__nth__set__encoding__conv,axiom,
    ! [L: list_P6011104703257516679at_nat] : ran_na2114640787166747904at_nat(aTP_Lamm_gk(L)) = set_Pr5648618587558075414at_nat(L) ).

% ran_nth_set_encoding_conv
tff(fact_2759_ran__nth__set__encoding__conv,axiom,
    ! [L: list_num] : ran_nat_num(aTP_Lamm_gl(L)) = set_num2(L) ).

% ran_nth_set_encoding_conv
tff(fact_2760_ran__nth__set__encoding__conv,axiom,
    ! [L: list_real] : ran_nat_real(aTP_Lamm_gm(L)) = aa_lis3769085826154824284t_real(set_real2,L) ).

% ran_nth_set_encoding_conv
tff(fact_2761_ran__nth__set__encoding__conv,axiom,
    ! [L: list_bool] : ran_nat_bool(aTP_Lamm_gn(L)) = aa_lis3587183946281582166t_bool(set_bool2,L) ).

% ran_nth_set_encoding_conv
tff(fact_2762_ran__nth__set__encoding__conv,axiom,
    ! [L: list_nat] : ran_nat_nat(aTP_Lamm_go(L)) = aa_list_nat_set_nat(set_nat2,L) ).

% ran_nth_set_encoding_conv
tff(fact_2763_ran__nth__set__encoding__conv,axiom,
    ! [L: list_int] : ran_nat_int(aTP_Lamm_gp(L)) = aa_list_int_set_int(set_int2,L) ).

% ran_nth_set_encoding_conv
tff(fact_2764_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,L)))
         => ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),L))
           => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(nth_bool(L),I)),X3))
             => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X3),aa_nat_bool(nth_bool(L),J)))
               => ~ ! [K2: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K2))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),J))
                       => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(nth_bool(L),K2)),X3))
                         => ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X3),aa_nat_bool(nth_bool(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_2765_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_real,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_real_nat(size_size_list_real,L)))
         => ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),L))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(nth_real(L),I)),X3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_nat_real(nth_real(L),J)))
               => ~ ! [K2: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K2))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),J))
                       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(nth_real(L),K2)),X3))
                         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),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_2766_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_rat,X3: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_rat(L)))
         => ( sorted_wrt_rat(ord_less_eq_rat,L)
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(nth_rat(L),I)),X3))
             => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),aa_nat_rat(nth_rat(L),J)))
               => ~ ! [K2: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K2))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),J))
                       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(nth_rat(L),K2)),X3))
                         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),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_2767_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_num,X3: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_num(L)))
         => ( sorted_wrt_num(ord_less_eq_num,L)
           => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(nth_num(L),I)),X3))
             => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),aa_nat_num(nth_num(L),J)))
               => ~ ! [K2: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K2))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),J))
                       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(nth_num(L),K2)),X3))
                         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),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_2768_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L)))
         => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(L),I)),X3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(nth_nat(L),J)))
               => ~ ! [K2: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K2))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),J))
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(L),K2)),X3))
                         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),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_2769_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_int,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(size_size_list_int,L)))
         => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(L),I)),X3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),aa_nat_int(nth_int(L),J)))
               => ~ ! [K2: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K2))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),J))
                       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(L),K2)),X3))
                         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),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_2770_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBTi,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_s7982070591426661849_VEBTi(Xs)))
     => ( nth_Pr3244165891152107629_VEBTi(enumerate_VEBT_VEBTi(N,Xs),M) = produc2649746096677893406_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2771_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(N,Xs),M) = aa_VEB2495835560080880209T_VEBT(produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2772_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_size_list_num(Xs)))
     => ( nth_Pr8326237132889035090at_num(enumerate_num(N,Xs),M) = aa_num5026351239867140595at_num(product_Pair_nat_num(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_num(nth_num(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2773_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_real,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_real_nat(size_size_list_real,Xs)))
     => ( nth_Pr7767817059697521252t_real(enumerate_real(N,Xs),M) = aa_rea4353191885927217907t_real(produc7837566107596912789t_real(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_real(nth_real(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2774_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( nth_Pr396318242284072693t_bool(enumerate_bool(N,Xs),M) = aa_boo8466755144949572503t_bool(produc3913204875133826030t_bool(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_bool(nth_bool(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2775_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(N,Xs),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_nat(nth_nat(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2776_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_int_nat(size_size_list_int,Xs)))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(N,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2777_map__ident,axiom,
    ! [X5: list_nat] : map_nat_nat(aTP_Lamm_dl,X5) = X5 ).

% map_ident
tff(fact_2778_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2779_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_2780_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2781_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_2782_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_2783_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),zero_zero_complex) = A ).

% diff_zero
tff(fact_2784_diff__zero,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_zero
tff(fact_2785_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_zero
tff(fact_2786_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_2787_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_2788_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_2789_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_2790_diff__0__right,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_0_right
tff(fact_2791_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_2792_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_2793_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),A) = zero_zero_complex ).

% diff_self
tff(fact_2794_diff__self,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% diff_self
tff(fact_2795_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% diff_self
tff(fact_2796_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_2797_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_2798_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_2799_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_2800_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_2801_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_right
tff(fact_2802_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_right
tff(fact_2803_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_2804_add__diff__cancel__right,axiom,
    ! [A: int,C: 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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_2805_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_2806_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_2807_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_2808_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_2809_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_left
tff(fact_2810_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_left
tff(fact_2811_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_2812_add__diff__cancel__left,axiom,
    ! [C: 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,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_2813_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_2814_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ).

% diff_add_cancel
tff(fact_2815_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_2816_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_2817_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel
tff(fact_2818_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_2819_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) ).

% Suc_diff_diff
tff(fact_2820_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_Suc_Suc
tff(fact_2821_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_2822_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_2823_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_2824_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_2825_length__map,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] : aa_list_real_nat(size_size_list_real,map_VEBT_VEBT_real(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_2826_length__map,axiom,
    ! [F: fun_real_real,Xs: list_real] : aa_list_real_nat(size_size_list_real,map_real_real(F,Xs)) = aa_list_real_nat(size_size_list_real,Xs) ).

% length_map
tff(fact_2827_length__map,axiom,
    ! [F: fun_bool_real,Xs: list_bool] : aa_list_real_nat(size_size_list_real,map_bool_real(F,Xs)) = aa_list_bool_nat(size_size_list_bool,Xs) ).

% length_map
tff(fact_2828_length__map,axiom,
    ! [F: fun_nat_real,Xs: list_nat] : aa_list_real_nat(size_size_list_real,map_nat_real(F,Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_map
tff(fact_2829_length__map,axiom,
    ! [F: fun_int_real,Xs: list_int] : aa_list_real_nat(size_size_list_real,map_int_real(F,Xs)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_map
tff(fact_2830_length__map,axiom,
    ! [F: fun_real_bool,Xs: list_real] : aa_list_bool_nat(size_size_list_bool,map_real_bool(F,Xs)) = aa_list_real_nat(size_size_list_real,Xs) ).

% length_map
tff(fact_2831_length__map,axiom,
    ! [F: fun_bool_bool,Xs: list_bool] : aa_list_bool_nat(size_size_list_bool,map_bool_bool(F,Xs)) = aa_list_bool_nat(size_size_list_bool,Xs) ).

% length_map
tff(fact_2832_length__map,axiom,
    ! [F: fun_nat_bool,Xs: list_nat] : aa_list_bool_nat(size_size_list_bool,map_nat_bool(F,Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_map
tff(fact_2833_length__map,axiom,
    ! [F: fun_int_bool,Xs: list_int] : aa_list_bool_nat(size_size_list_bool,map_int_bool(F,Xs)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_map
tff(fact_2834_length__map,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] : aa_list_nat_nat(size_size_list_nat,map_VEBT_VEBT_nat(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_2835_length__enumerate,axiom,
    ! [N: nat,Xs: list_real] : size_s7910714270633306959t_real(enumerate_real(N,Xs)) = aa_list_real_nat(size_size_list_real,Xs) ).

% length_enumerate
tff(fact_2836_length__enumerate,axiom,
    ! [N: nat,Xs: list_bool] : size_s783543119881238752t_bool(enumerate_bool(N,Xs)) = aa_list_bool_nat(size_size_list_bool,Xs) ).

% length_enumerate
tff(fact_2837_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(N,Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_enumerate
tff(fact_2838_length__enumerate,axiom,
    ! [N: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(N,Xs)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_enumerate
tff(fact_2839_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_2840_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_2841_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_2842_zero__comp__diff__simps_I1_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% zero_comp_diff_simps(1)
tff(fact_2843_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% zero_comp_diff_simps(1)
tff(fact_2844_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% zero_comp_diff_simps(1)
tff(fact_2845_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_2846_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_2847_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_2848_zero__comp__diff__simps_I2_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% zero_comp_diff_simps(2)
tff(fact_2849_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% zero_comp_diff_simps(2)
tff(fact_2850_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% zero_comp_diff_simps(2)
tff(fact_2851_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_2852_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_2853_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_2854_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_2855_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_2856_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2857_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2858_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2859_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_2860_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_2861_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2862_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2863_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_2864_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% zero_less_diff
tff(fact_2865_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_2866_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_2867_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% diff_is_0_eq
tff(fact_2868_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_2869_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_2870_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_2871_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_2872_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_2873_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_2874_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_2875_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBTi_nat(F,Xs)),N) = aa_VEBT_VEBTi_nat(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N)) ) ) ).

% nth_map
tff(fact_2876_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: fun_VE8449866691444886503_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(map_VE483055756984248624_VEBTi(F,Xs)),N) = aa_VEB5586453056816421800_VEBTi(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N)) ) ) ).

% nth_map
tff(fact_2877_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VE712148701811721852_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(map_VE7029150624388687525_VEBTi(F,Xs)),N) = aa_VEB3973023575446460973_VEBTi(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)) ) ) ).

% nth_map
tff(fact_2878_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: fun_VE9120655849212850072T_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_VE7998069337340375161T_VEBT(F,Xs)),N) = aa_VEB4941942288398148609T_VEBT(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N)) ) ) ).

% nth_map
tff(fact_2879_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_VE8901447254227204932T_VEBT(F,Xs)),N) = aa_VEB5832953974749077948T_VEBT(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)) ) ) ).

% nth_map
tff(fact_2880_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( aa_nat_int(nth_int(map_VEBT_VEBTi_int(F,Xs)),N) = aa_VEBT_VEBTi_int(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N)) ) ) ).

% nth_map
tff(fact_2881_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_int(nth_int(map_VEBT_VEBT_int(F,Xs)),N) = aa_VEBT_VEBT_int(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)) ) ) ).

% nth_map
tff(fact_2882_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBT_nat(F,Xs)),N) = aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)) ) ) ).

% nth_map
tff(fact_2883_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_real(nth_real(map_VEBT_VEBT_real(F,Xs)),N) = aa_VEBT_VEBT_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)) ) ) ).

% nth_map
tff(fact_2884_nth__map,axiom,
    ! [N: nat,Xs: list_real,F: fun_real_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
     => ( aa_nat_nat(nth_nat(map_real_nat(F,Xs)),N) = aa_real_nat(F,aa_nat_real(nth_real(Xs),N)) ) ) ).

% nth_map
tff(fact_2885_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_2886_sorted__map,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),map_VEBT_VEBT_real(F,Xs)))
    <=> sorted_wrt_VEBT_VEBT(aTP_Lamm_gq(F),Xs) ) ).

% sorted_map
tff(fact_2887_sorted__map,axiom,
    ! [F: fun_nat_bool,Xs: list_nat] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),map_nat_bool(F,Xs)))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(aTP_Lamm_gr(F)),Xs)) ) ).

% sorted_map
tff(fact_2888_sorted__map,axiom,
    ! [F: fun_nat_rat,Xs: list_nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,map_nat_rat(F,Xs))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(aTP_Lamm_gs(F)),Xs)) ) ).

% sorted_map
tff(fact_2889_sorted__map,axiom,
    ! [F: fun_int_rat,Xs: list_int] :
      ( sorted_wrt_rat(ord_less_eq_rat,map_int_rat(F,Xs))
    <=> pp(aa_list_int_bool(sorted_wrt_int(aTP_Lamm_gt(F)),Xs)) ) ).

% sorted_map
tff(fact_2890_sorted__map,axiom,
    ! [F: fun_nat_num,Xs: list_nat] :
      ( sorted_wrt_num(ord_less_eq_num,map_nat_num(F,Xs))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(aTP_Lamm_gu(F)),Xs)) ) ).

% sorted_map
tff(fact_2891_sorted__map,axiom,
    ! [F: fun_int_num,Xs: list_int] :
      ( sorted_wrt_num(ord_less_eq_num,map_int_num(F,Xs))
    <=> pp(aa_list_int_bool(sorted_wrt_int(aTP_Lamm_gv(F)),Xs)) ) ).

% sorted_map
tff(fact_2892_sorted__map,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_VEBT_VEBT_nat(F,Xs)))
    <=> sorted_wrt_VEBT_VEBT(aTP_Lamm_gw(F),Xs) ) ).

% sorted_map
tff(fact_2893_sorted__map,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_nat_nat(F,Xs)))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(aTP_Lamm_gx(F)),Xs)) ) ).

% sorted_map
tff(fact_2894_sorted__map,axiom,
    ! [F: fun_int_nat,Xs: list_int] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_int_nat(F,Xs)))
    <=> pp(aa_list_int_bool(sorted_wrt_int(aTP_Lamm_gy(F)),Xs)) ) ).

% sorted_map
tff(fact_2895_sorted__map,axiom,
    ! [F: fun_nat_int,Xs: list_nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),map_nat_int(F,Xs)))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(aTP_Lamm_gz(F)),Xs)) ) ).

% sorted_map
tff(fact_2896_sorted__wrt__true,axiom,
    ! [Xs: list_nat] : pp(aa_list_nat_bool(sorted_wrt_nat(aTP_Lamm_ha),Xs)) ).

% sorted_wrt_true
tff(fact_2897_sorted__wrt__true,axiom,
    ! [Xs: list_int] : pp(aa_list_int_bool(sorted_wrt_int(aTP_Lamm_hb),Xs)) ).

% sorted_wrt_true
tff(fact_2898_sorted__wrt__map,axiom,
    ! [R: fun_re1255401033344146673l_bool,F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] :
      ( pp(aa_list_real_bool(sorted_wrt_real(R),map_VEBT_VEBT_real(F,Xs)))
    <=> sorted_wrt_VEBT_VEBT(aa_fun2912881648665429622T_bool(aTP_Lamm_hc(R),F),Xs) ) ).

% sorted_wrt_map
tff(fact_2899_sorted__wrt__map,axiom,
    ! [R: fun_bo1841576699660343469l_bool,F: fun_Pr4298194452408323251l_bool,Xs: list_P2722043638750504429l_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(R),map_Pr8574941368599290932l_bool(F,Xs)))
    <=> sorted378674801997687508l_bool(aa_fun4430674632827408636l_bool(aTP_Lamm_hd(R),F),Xs) ) ).

% sorted_wrt_map
tff(fact_2900_sorted__wrt__map,axiom,
    ! [R: fun_bo1841576699660343469l_bool,F: fun_nat_bool,Xs: list_nat] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(R),map_nat_bool(F,Xs)))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(aa_fun8271680885375493061t_bool(aTP_Lamm_he(R),F)),Xs)) ) ).

% sorted_wrt_map
tff(fact_2901_sorted__wrt__map,axiom,
    ! [R: fun_nat_fun_nat_bool,F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(R),map_VEBT_VEBT_nat(F,Xs)))
    <=> sorted_wrt_VEBT_VEBT(aa_fun7053968375109723858T_bool(aTP_Lamm_hf(R),F),Xs) ) ).

% sorted_wrt_map
tff(fact_2902_sorted__wrt__map,axiom,
    ! [R: fun_nat_fun_nat_bool,F: fun_nat_nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(R),map_nat_nat(F,Xs)))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(aa_fun3777555276967886990t_bool(aTP_Lamm_hg(R),F)),Xs)) ) ).

% sorted_wrt_map
tff(fact_2903_sorted__wrt__map,axiom,
    ! [R: fun_nat_fun_nat_bool,F: fun_int_nat,Xs: list_int] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(R),map_int_nat(F,Xs)))
    <=> pp(aa_list_int_bool(sorted_wrt_int(aa_fun2912556895731619962t_bool(aTP_Lamm_hh(R),F)),Xs)) ) ).

% sorted_wrt_map
tff(fact_2904_sorted__wrt__map,axiom,
    ! [R: fun_int_fun_int_bool,F: fun_nat_int,Xs: list_nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(R),map_nat_int(F,Xs)))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(aa_fun2605284095902351154t_bool(aTP_Lamm_hi(R),F)),Xs)) ) ).

% sorted_wrt_map
tff(fact_2905_sorted__wrt__map,axiom,
    ! [R: fun_int_fun_int_bool,F: fun_int_int,Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(R),map_int_int(F,Xs)))
    <=> pp(aa_list_int_bool(sorted_wrt_int(aa_fun1740285714666084126t_bool(aTP_Lamm_hj(R),F)),Xs)) ) ).

% sorted_wrt_map
tff(fact_2906_list_Omap__ident,axiom,
    ! [T: list_nat] : map_nat_nat(aTP_Lamm_dl,T) = T ).

% list.map_ident
tff(fact_2907_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_2908_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2909_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2910_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2911_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: 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),C)),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)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2912_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_2913_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_2914_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: 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,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_2915_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2916_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2917_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_real_nat,Ys: list_real] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_real_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = aa_list_real_nat(size_size_list_real,Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2918_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_real_real,Ys: list_real] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_real_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = aa_list_real_nat(size_size_list_real,Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2919_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_bool_nat,Ys: list_bool] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_bool_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = aa_list_bool_nat(size_size_list_bool,Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2920_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_bool_real,Ys: list_bool] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_bool_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = aa_list_bool_nat(size_size_list_bool,Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2921_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_nat_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2922_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_nat_real,Ys: list_nat] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_nat_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2923_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_int_nat,Ys: list_int] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_int_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = aa_list_int_nat(size_size_list_int,Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2924_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_int_real,Ys: list_int] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_int_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = aa_list_int_nat(size_size_list_int,Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2925_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_2926_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_2927_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_2928_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_2929_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2930_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2931_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: 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,C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2932_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C))) ) ).

% diff_right_mono
tff(fact_2933_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C))) ) ).

% diff_right_mono
tff(fact_2934_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_right_mono
tff(fact_2935_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B))) ) ).

% diff_left_mono
tff(fact_2936_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))) ) ).

% diff_left_mono
tff(fact_2937_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_left_mono
tff(fact_2938_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D))) ) ) ).

% diff_mono
tff(fact_2939_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D))) ) ) ).

% diff_mono
tff(fact_2940_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_mono
tff(fact_2941_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C))) ) ).

% diff_strict_right_mono
tff(fact_2942_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C))) ) ).

% diff_strict_right_mono
tff(fact_2943_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_strict_right_mono
tff(fact_2944_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B))) ) ).

% diff_strict_left_mono
tff(fact_2945_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))) ) ).

% diff_strict_left_mono
tff(fact_2946_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_strict_left_mono
tff(fact_2947_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_2948_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_2949_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: 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,C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_2950_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D))) ) ) ).

% diff_strict_mono
tff(fact_2951_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D))) ) ) ).

% diff_strict_mono
tff(fact_2952_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_strict_mono
tff(fact_2953_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_2954_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_2955_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_2956_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_2957_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(minus_minus_real(A),B) ) ) ).

% add_implies_diff
tff(fact_2958_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
     => ( C = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ) ) ).

% add_implies_diff
tff(fact_2959_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_2960_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_2961_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_2962_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_2963_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_2964_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_2965_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_add_eq
tff(fact_2966_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_2967_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_2968_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_diff_eq2
tff(fact_2969_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_2970_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_2971_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ).

% add_diff_eq
tff(fact_2972_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_2973_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(minus_minus_real(C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_2974_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_2975_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_2976_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_2977_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_2978_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_2979_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(minus_minus_real(A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_2980_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_2981_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),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_2982_inf__period_I1_J,axiom,
    ! [P: fun_real_bool,D4: real,Q: fun_real_bool] :
      ( ! [X: real,K2: real] :
          ( pp(aa_real_bool(P,X))
        <=> pp(aa_real_bool(P,aa_real_real(minus_minus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
     => ( ! [X: real,K2: real] :
            ( pp(aa_real_bool(Q,X))
          <=> pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
       => ! [X5: real,K4: real] :
            ( ( pp(aa_real_bool(P,X5))
              & pp(aa_real_bool(Q,X5)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))))
              & pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2983_inf__period_I1_J,axiom,
    ! [P: fun_rat_bool,D4: rat,Q: fun_rat_bool] :
      ( ! [X: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X))
        <=> pp(aa_rat_bool(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D4)))) )
     => ( ! [X: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D4)))) )
       => ! [X5: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X5))
              & pp(aa_rat_bool(Q,X5)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))))
              & pp(aa_rat_bool(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2984_inf__period_I1_J,axiom,
    ! [P: fun_int_bool,D4: int,Q: fun_int_bool] :
      ( ! [X: int,K2: int] :
          ( pp(aa_int_bool(P,X))
        <=> pp(aa_int_bool(P,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: int,K2: int] :
            ( pp(aa_int_bool(Q,X))
          <=> pp(aa_int_bool(Q,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)))) )
       => ! [X5: int,K4: int] :
            ( ( pp(aa_int_bool(P,X5))
              & pp(aa_int_bool(Q,X5)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))))
              & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2985_inf__period_I1_J,axiom,
    ! [P: fun_complex_bool,D4: complex,Q: fun_complex_bool] :
      ( ! [X: complex,K2: complex] :
          ( pp(aa_complex_bool(P,X))
        <=> pp(aa_complex_bool(P,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,K2),D4)))) )
     => ( ! [X: complex,K2: complex] :
            ( pp(aa_complex_bool(Q,X))
          <=> pp(aa_complex_bool(Q,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,K2),D4)))) )
       => ! [X5: complex,K4: complex] :
            ( ( pp(aa_complex_bool(P,X5))
              & pp(aa_complex_bool(Q,X5)) )
          <=> ( pp(aa_complex_bool(P,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X5),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,K4),D4))))
              & pp(aa_complex_bool(Q,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X5),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,K4),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_2986_inf__period_I2_J,axiom,
    ! [P: fun_real_bool,D4: real,Q: fun_real_bool] :
      ( ! [X: real,K2: real] :
          ( pp(aa_real_bool(P,X))
        <=> pp(aa_real_bool(P,aa_real_real(minus_minus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
     => ( ! [X: real,K2: real] :
            ( pp(aa_real_bool(Q,X))
          <=> pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
       => ! [X5: real,K4: real] :
            ( ( pp(aa_real_bool(P,X5))
              | pp(aa_real_bool(Q,X5)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))))
              | pp(aa_real_bool(Q,aa_real_real(minus_minus_real(X5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2987_inf__period_I2_J,axiom,
    ! [P: fun_rat_bool,D4: rat,Q: fun_rat_bool] :
      ( ! [X: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X))
        <=> pp(aa_rat_bool(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D4)))) )
     => ( ! [X: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D4)))) )
       => ! [X5: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X5))
              | pp(aa_rat_bool(Q,X5)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))))
              | pp(aa_rat_bool(Q,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2988_inf__period_I2_J,axiom,
    ! [P: fun_int_bool,D4: int,Q: fun_int_bool] :
      ( ! [X: int,K2: int] :
          ( pp(aa_int_bool(P,X))
        <=> pp(aa_int_bool(P,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: int,K2: int] :
            ( pp(aa_int_bool(Q,X))
          <=> pp(aa_int_bool(Q,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)))) )
       => ! [X5: int,K4: int] :
            ( ( pp(aa_int_bool(P,X5))
              | pp(aa_int_bool(Q,X5)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))))
              | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2989_inf__period_I2_J,axiom,
    ! [P: fun_complex_bool,D4: complex,Q: fun_complex_bool] :
      ( ! [X: complex,K2: complex] :
          ( pp(aa_complex_bool(P,X))
        <=> pp(aa_complex_bool(P,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,K2),D4)))) )
     => ( ! [X: complex,K2: complex] :
            ( pp(aa_complex_bool(Q,X))
          <=> pp(aa_complex_bool(Q,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,K2),D4)))) )
       => ! [X5: complex,K4: complex] :
            ( ( pp(aa_complex_bool(P,X5))
              | pp(aa_complex_bool(Q,X5)) )
          <=> ( pp(aa_complex_bool(P,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X5),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,K4),D4))))
              | pp(aa_complex_bool(Q,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X5),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,K4),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_2990_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_2991_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib'
tff(fact_2992_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_2993_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_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,A),C)) ).

% right_diff_distrib'
tff(fact_2994_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)) ).

% right_diff_distrib'
tff(fact_2995_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),C)),A) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_2996_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)) ).

% left_diff_distrib'
tff(fact_2997_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_2998_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_2999_left__diff__distrib_H,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),C)),A) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A)) ).

% left_diff_distrib'
tff(fact_3000_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_3001_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib
tff(fact_3002_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_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,A),C)) ).

% right_diff_distrib
tff(fact_3003_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)) ).

% right_diff_distrib
tff(fact_3004_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_3005_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% left_diff_distrib
tff(fact_3006_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_3007_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) ).

% left_diff_distrib
tff(fact_3008_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_3009_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_3010_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_3011_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_3012_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).

% diff_less_mono2
tff(fact_3013_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).

% diff_add_inverse2
tff(fact_3014_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).

% diff_add_inverse
tff(fact_3015_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_cancel2
tff(fact_3016_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% Nat.diff_cancel
tff(fact_3017_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).

% diff_le_mono2
tff(fact_3018_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).

% le_diff_iff'
tff(fact_3019_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).

% diff_le_self
tff(fact_3020_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).

% diff_le_mono
tff(fact_3021_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_3022_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% le_diff_iff
tff(fact_3023_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_3024_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% diff_mult_distrib2
tff(fact_3025_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% diff_mult_distrib
tff(fact_3026_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_real),Xs))
     => pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs)) ) ).

% strict_sorted_imp_sorted
tff(fact_3027_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_3028_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_3029_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Xs))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs)) ) ).

% strict_sorted_imp_sorted
tff(fact_3030_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),Xs))
     => pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs)) ) ).

% strict_sorted_imp_sorted
tff(fact_3031_strict__sorted__equal,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_bool),Xs))
     => ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_bool),Ys))
       => ( ( aa_lis3587183946281582166t_bool(set_bool2,Ys) = aa_lis3587183946281582166t_bool(set_bool2,Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_3032_strict__sorted__equal,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_real),Xs))
     => ( pp(aa_list_real_bool(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_3033_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_3034_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_3035_strict__sorted__equal,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Xs))
     => ( pp(aa_list_nat_bool(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_3036_strict__sorted__equal,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),Xs))
     => ( pp(aa_list_int_bool(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_3037_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_3038_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_3039_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_3040_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_3041_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_3042_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(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_3043_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),aa_real_real(minus_minus_real(N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3044_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3045_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3046_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3047_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3048_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3049_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3050_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3051_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_le_eq
tff(fact_3052_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% diff_le_eq
tff(fact_3053_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_le_eq
tff(fact_3054_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(minus_minus_real(C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% le_diff_eq
tff(fact_3055_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C)) ) ).

% le_diff_eq
tff(fact_3056_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% le_diff_eq
tff(fact_3057_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3058_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A))) ) ).

% le_add_diff
tff(fact_3059_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3060_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3061_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3062_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3063_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3064_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3065_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3066_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3067_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_3068_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3069_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3070_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(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_3071_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_less_eq
tff(fact_3072_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))) ) ).

% diff_less_eq
tff(fact_3073_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_less_eq
tff(fact_3074_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(minus_minus_real(C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% less_diff_eq
tff(fact_3075_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C)) ) ).

% less_diff_eq
tff(fact_3076_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% less_diff_eq
tff(fact_3077_square__diff__square__factored,axiom,
    ! [X3: real,Y: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y)),aa_real_real(minus_minus_real(X3),Y)) ).

% square_diff_square_factored
tff(fact_3078_square__diff__square__factored,axiom,
    ! [X3: rat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),Y)) ).

% square_diff_square_factored
tff(fact_3079_square__diff__square__factored,axiom,
    ! [X3: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)) ).

% square_diff_square_factored
tff(fact_3080_square__diff__square__factored,axiom,
    ! [X3: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),X3)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X3),Y)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X3),Y)) ).

% square_diff_square_factored
tff(fact_3081_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3082_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3083_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = 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) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3084_eq__add__iff2,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),E2)),C) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),E2)),D) )
    <=> ( C = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3085_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3086_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3087_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = 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_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)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3088_eq__add__iff1,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),E2)),C) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),E2)),D) )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3089_mult__diff__mult,axiom,
    ! [X3: real,Y: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),aa_real_real(minus_minus_real(Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(X3),A)),B)) ).

% mult_diff_mult
tff(fact_3090_mult__diff__mult,axiom,
    ! [X3: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),A)),B)) ).

% mult_diff_mult
tff(fact_3091_mult__diff__mult,axiom,
    ! [X3: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),A)),B)) ).

% mult_diff_mult
tff(fact_3092_mult__diff__mult,axiom,
    ! [X3: complex,Y: complex,A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),Y)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y),B))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X3),A)),B)) ).

% mult_diff_mult
tff(fact_3093_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(suc,N) = M )
     => ( N = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_3094_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_3095_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(suc,M))) ).

% diff_less_Suc
tff(fact_3096_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).

% diff_less
tff(fact_3097_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_3098_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_3099_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_3100_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ).

% less_diff_conv
tff(fact_3101_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_3102_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C))) ) ) ).

% diff_less_mono
tff(fact_3103_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ).

% less_diff_iff
tff(fact_3104_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).

% le_diff_conv
tff(fact_3105_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_3106_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_3107_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_3108_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_3109_sorted__wrt01,axiom,
    ! [Xs: list_real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),one_one_nat))
     => pp(aa_list_real_bool(sorted_wrt_real(P),Xs)) ) ).

% sorted_wrt01
tff(fact_3110_sorted__wrt01,axiom,
    ! [Xs: list_bool,P: fun_bo1841576699660343469l_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),one_one_nat))
     => pp(aa_list_bool_bool(sorted_wrt_bool(P),Xs)) ) ).

% sorted_wrt01
tff(fact_3111_sorted__wrt01,axiom,
    ! [Xs: list_nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat))
     => pp(aa_list_nat_bool(sorted_wrt_nat(P),Xs)) ) ).

% sorted_wrt01
tff(fact_3112_sorted__wrt01,axiom,
    ! [Xs: list_int,P: fun_int_fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),one_one_nat))
     => pp(aa_list_int_bool(sorted_wrt_int(P),Xs)) ) ).

% sorted_wrt01
tff(fact_3113_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_VE6376819126223186655i_bool,Xs: list_VEBT_VEBTi] :
      ( sorted9206477368072086664_VEBTi(P,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),size_s7982070591426661849_VEBTi(Xs)))
           => pp(aa_VEBT_VEBTi_bool(aa_VEB8608361877424424648i_bool(P,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_3114_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_VE5877677516038742509T_bool,Xs: list_VEBT_VEBT] :
      ( sorted_wrt_VEBT_VEBT(P,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),size_s6755466524823107622T_VEBT(Xs)))
           => pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(P,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_3115_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_re1255401033344146673l_bool,Xs: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(P),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_real_nat(size_size_list_real,Xs)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_nat_real(nth_real(Xs),I3)),aa_nat_real(nth_real(Xs),J3))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3116_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_bo1841576699660343469l_bool,Xs: list_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(P),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_bool_nat(size_size_list_bool,Xs)))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(P,aa_nat_bool(nth_bool(Xs),I3)),aa_nat_bool(nth_bool(Xs),J3))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3117_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_nat_fun_nat_bool,Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(P),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Xs),J3))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3118_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_int_fun_int_bool,Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(P),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_int_nat(size_size_list_int,Xs)))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_nat_int(nth_int(Xs),I3)),aa_nat_int(nth_int(Xs),J3))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3119_sorted__wrt__nth__less,axiom,
    ! [P: fun_VE6376819126223186655i_bool,Xs: list_VEBT_VEBTi,I: nat,J: nat] :
      ( sorted9206477368072086664_VEBTi(P,Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s7982070591426661849_VEBTi(Xs)))
         => pp(aa_VEBT_VEBTi_bool(aa_VEB8608361877424424648i_bool(P,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3120_sorted__wrt__nth__less,axiom,
    ! [P: fun_VE5877677516038742509T_bool,Xs: list_VEBT_VEBT,I: nat,J: nat] :
      ( sorted_wrt_VEBT_VEBT(P,Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3121_sorted__wrt__nth__less,axiom,
    ! [P: fun_re1255401033344146673l_bool,Xs: list_real,I: nat,J: nat] :
      ( pp(aa_list_real_bool(sorted_wrt_real(P),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_real_nat(size_size_list_real,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3122_sorted__wrt__nth__less,axiom,
    ! [P: fun_bo1841576699660343469l_bool,Xs: list_bool,I: nat,J: nat] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(P),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,Xs)))
         => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(P,aa_nat_bool(nth_bool(Xs),I)),aa_nat_bool(nth_bool(Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3123_sorted__wrt__nth__less,axiom,
    ! [P: fun_nat_fun_nat_bool,Xs: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(P),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3124_sorted__wrt__nth__less,axiom,
    ! [P: fun_int_fun_int_bool,Xs: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(P),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(size_size_list_int,Xs)))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3125_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3126_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3127_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3128_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3129_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3130_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3131_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3132_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3133_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_3134_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3135_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3136_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),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)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_3137_square__diff__one__factored,axiom,
    ! [X3: real] : aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),one_one_real)),aa_real_real(minus_minus_real(X3),one_one_real)) ).

% square_diff_one_factored
tff(fact_3138_square__diff__one__factored,axiom,
    ! [X3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),one_one_rat)) ).

% square_diff_one_factored
tff(fact_3139_square__diff__one__factored,axiom,
    ! [X3: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),one_one_int)) ).

% square_diff_one_factored
tff(fact_3140_square__diff__one__factored,axiom,
    ! [X3: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),X3)),one_one_complex) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X3),one_one_complex)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,X3),one_one_complex)) ).

% square_diff_one_factored
tff(fact_3141_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_3142_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_3143_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ! [D5: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
           => pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split
tff(fact_3144_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) )
          | ? [D5: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
              & ~ pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split_asm
tff(fact_3145_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).

% less_diff_conv2
tff(fact_3146_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_3147_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_3148_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_3149_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_3150_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_3151_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_3152_diff__preserves__multiset,axiom,
    ! [M8: fun_list_nat_nat,N7: fun_list_nat_nat] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_bn(M8)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun2581403542315806821t_bool(aTP_Lamm_hk(M8),N7))) ) ).

% diff_preserves_multiset
tff(fact_3153_diff__preserves__multiset,axiom,
    ! [M8: fun_set_nat_nat,N7: fun_set_nat_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_bp(M8))))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_fun806690937814068505t_bool(aTP_Lamm_hl(M8),N7)))) ) ).

% diff_preserves_multiset
tff(fact_3154_diff__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,N7: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_br(M8))))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aTP_Lamm_hm(M8),N7)))) ) ).

% diff_preserves_multiset
tff(fact_3155_diff__preserves__multiset,axiom,
    ! [M8: fun_int_nat,N7: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_bt(M8))))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5627573752281863885t_bool(aTP_Lamm_hn(M8),N7)))) ) ).

% diff_preserves_multiset
tff(fact_3156_diff__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,N7: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_bv(M8))))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun4400319914713427657x_bool(aTP_Lamm_ho(M8),N7)))) ) ).

% diff_preserves_multiset
tff(fact_3157_diff__preserves__multiset,axiom,
    ! [M8: fun_Code_integer_nat,N7: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_bx(M8))))
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun3540204951715082475r_bool(aTP_Lamm_hp(M8),N7)))) ) ).

% diff_preserves_multiset
tff(fact_3158_sorted01,axiom,
    ! [Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),one_one_nat))
     => pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs)) ) ).

% sorted01
tff(fact_3159_sorted01,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),one_one_nat))
     => pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs)) ) ).

% sorted01
tff(fact_3160_sorted01,axiom,
    ! [Xs: list_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_rat(Xs)),one_one_nat))
     => sorted_wrt_rat(ord_less_eq_rat,Xs) ) ).

% sorted01
tff(fact_3161_sorted01,axiom,
    ! [Xs: list_num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_num(Xs)),one_one_nat))
     => sorted_wrt_num(ord_less_eq_num,Xs) ) ).

% sorted01
tff(fact_3162_sorted01,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs)) ) ).

% sorted01
tff(fact_3163_sorted01,axiom,
    ! [Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),one_one_nat))
     => pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs)) ) ).

% sorted01
tff(fact_3164_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_real_nat(size_size_list_real,Xs)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_3165_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_bool_nat(size_size_list_bool,Xs)))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(nth_bool(Xs),I3)),aa_nat_bool(nth_bool(Xs),J3))) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_3166_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),size_size_list_rat(Xs)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3167_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),size_size_list_num(Xs)))
           => pp(aa_num_bool(aa_num_fun_num_bool(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_3168_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3169_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_int_nat(size_size_list_int,Xs)))
           => pp(aa_int_bool(aa_int_fun_int_bool(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_3170_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3171_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3172_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_3173_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_3174_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_3175_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_3176_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_3177_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_3178_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I3)),aa_list_real_nat(size_size_list_real,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_3179_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I3)),aa_list_bool_nat(size_size_list_bool,Xs)))
         => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(nth_bool(Xs),I3)),aa_nat_bool(nth_bool(Xs),aa_nat_nat(suc,I3)))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_3180_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I3)),size_size_list_rat(Xs)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(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_3181_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I3)),size_size_list_num(Xs)))
         => pp(aa_num_bool(aa_num_fun_num_bool(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_3182_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I3)),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3183_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I3)),aa_list_int_nat(size_size_list_int,Xs)))
         => pp(aa_int_bool(aa_int_fun_int_bool(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_3184_sorted__nth__mono,axiom,
    ! [Xs: list_real,I: nat,J: nat] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_real_nat(size_size_list_real,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xs),J))) ) ) ) ).

% sorted_nth_mono
tff(fact_3185_sorted__nth__mono,axiom,
    ! [Xs: list_bool,I: nat,J: nat] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,Xs)))
         => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(nth_bool(Xs),I)),aa_nat_bool(nth_bool(Xs),J))) ) ) ) ).

% sorted_nth_mono
tff(fact_3186_sorted__nth__mono,axiom,
    ! [Xs: list_rat,I: nat,J: nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_rat(Xs)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I)),aa_nat_rat(nth_rat(Xs),J))) ) ) ) ).

% sorted_nth_mono
tff(fact_3187_sorted__nth__mono,axiom,
    ! [Xs: list_num,I: nat,J: nat] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_num(Xs)))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(nth_num(Xs),I)),aa_nat_num(nth_num(Xs),J))) ) ) ) ).

% sorted_nth_mono
tff(fact_3188_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J))) ) ) ) ).

% sorted_nth_mono
tff(fact_3189_sorted__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(size_size_list_int,Xs)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J))) ) ) ) ).

% sorted_nth_mono
tff(fact_3190_sorted__iff__nth__mono,axiom,
    ! [Xs: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_real_nat(size_size_list_real,Xs)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(nth_real(Xs),I3)),aa_nat_real(nth_real(Xs),J3))) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3191_sorted__iff__nth__mono,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_bool_nat(size_size_list_bool,Xs)))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(nth_bool(Xs),I3)),aa_nat_bool(nth_bool(Xs),J3))) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3192_sorted__iff__nth__mono,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),size_size_list_rat(Xs)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I3)),aa_nat_rat(nth_rat(Xs),J3))) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3193_sorted__iff__nth__mono,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),size_size_list_num(Xs)))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(nth_num(Xs),I3)),aa_nat_num(nth_num(Xs),J3))) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3194_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Xs),J3))) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3195_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J3))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_int_nat(size_size_list_int,Xs)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(Xs),I3)),aa_nat_int(nth_int(Xs),J3))) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3196_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X22: bool] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,aa_bool_VEBT_VEBT(vEBT_Leaf(X21),X22)) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_3197_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_real,A2: fun_re6303261851654292474l_assn] :
      ( ( aa_list_real_nat(size_size_list_real,L) != aa_list_real_nat(size_size_list_real,Li2) )
     => ( vEBT_L1930518968523514909l_real(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3198_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_bool,A2: fun_re2206215847901373695l_assn] :
      ( ( aa_list_real_nat(size_size_list_real,L) != aa_list_bool_nat(size_size_list_bool,Li2) )
     => ( vEBT_L870199270341830246l_bool(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3199_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_nat,A2: fun_re574865750389657758t_assn] :
      ( ( aa_list_real_nat(size_size_list_real,L) != aa_list_nat_nat(size_size_list_nat,Li2) )
     => ( vEBT_L1446010312343316929al_nat(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3200_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_int,A2: fun_re4990660043736809722t_assn] :
      ( ( aa_list_real_nat(size_size_list_real,L) != aa_list_int_nat(size_size_list_int,Li2) )
     => ( vEBT_L1443519841834266653al_int(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3201_list__assn__aux__ineq__len,axiom,
    ! [L: list_bool,Li2: list_real,A2: fun_bo8885878543415060347l_assn] :
      ( ( aa_list_bool_nat(size_size_list_bool,L) != aa_list_real_nat(size_size_list_real,Li2) )
     => ( vEBT_L5107238702033023424l_real(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3202_list__assn__aux__ineq__len,axiom,
    ! [L: list_bool,Li2: list_bool,A2: fun_bo8172996160782635326l_assn] :
      ( ( aa_list_bool_nat(size_size_list_bool,L) != aa_list_bool_nat(size_size_list_bool,Li2) )
     => ( aa_list_bool_assn(vEBT_L6562251693209944067l_bool(A2,L),Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3203_list__assn__aux__ineq__len,axiom,
    ! [L: list_bool,Li2: list_nat,A2: fun_bo8438563643002758303t_assn] :
      ( ( aa_list_bool_nat(size_size_list_bool,L) != aa_list_nat_nat(size_size_list_nat,Li2) )
     => ( aa_list_nat_assn(vEBT_L7520539408223738084ol_nat(A2,L),Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3204_list__assn__aux__ineq__len,axiom,
    ! [L: list_bool,Li2: list_int,A2: fun_bo3630985899495134459t_assn] :
      ( ( aa_list_bool_nat(size_size_list_bool,L) != aa_list_int_nat(size_size_list_int,Li2) )
     => ( vEBT_L7518048937714687808ol_int(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3205_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li2: list_real,A2: fun_na448437756743965086l_assn] :
      ( ( aa_list_nat_nat(size_size_list_nat,L) != aa_list_real_nat(size_size_list_real,Li2) )
     => ( vEBT_L6102073776069194049t_real(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3206_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li2: list_bool,A2: fun_na5136419689362302555l_assn] :
      ( ( aa_list_nat_nat(size_size_list_nat,L) != aa_list_bool_nat(size_size_list_bool,Li2) )
     => ( vEBT_L2559740421116301378t_bool(A2,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_3207_repli__emp,axiom,
    ! [X3: heap_T2636463487746394924on_nat,A2: fun_VE6334927249922962060t_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_7629718768684598413on_nat(one_one_assn,X3,aa_VEB931861714549014589t_assn(A2,Y))
     => hoare_6480275734082232733on_nat(one_one_assn,vEBT_V792416675989592002on_nat(N,X3),vEBT_L8010285020845282001on_nat(A2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),Y))) ) ).

% repli_emp
tff(fact_3208_repli__emp,axiom,
    ! [X3: heap_T2636463487746394924on_nat,A2: fun_bo1795969981296709615t_assn,Y: bool,N: nat] :
      ( hoare_7629718768684598413on_nat(one_one_assn,X3,aa_boo2787173234663865760t_assn(A2,Y))
     => hoare_6480275734082232733on_nat(one_one_assn,vEBT_V792416675989592002on_nat(N,X3),vEBT_L5912725412549479476on_nat(A2,replicate_bool(N,Y))) ) ).

% repli_emp
tff(fact_3209_repli__emp,axiom,
    ! [X3: heap_Time_Heap_bool,A2: fun_VE1323945509527881313l_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_2774773592644289642e_bool(one_one_assn,X3,aa_VEB2879522196611654106l_assn(A2,Y))
     => hoare_9026296798553663728t_bool(one_one_assn,vEBT_V6044891547498605301i_bool(N,X3),vEBT_L5320321625355440422T_bool(A2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),Y))) ) ).

% repli_emp
tff(fact_3210_repli__emp,axiom,
    ! [X3: heap_Time_Heap_bool,A2: fun_bo8172996160782635326l_assn,Y: bool,N: nat] :
      ( hoare_2774773592644289642e_bool(one_one_assn,X3,aa_boo2938360491098410167l_assn(A2,Y))
     => hoare_9026296798553663728t_bool(one_one_assn,vEBT_V6044891547498605301i_bool(N,X3),vEBT_L6562251693209944067l_bool(A2,replicate_bool(N,Y))) ) ).

% repli_emp
tff(fact_3211_repli__emp,axiom,
    ! [X3: heap_T8145700208782473153_VEBTi,A2: fun_bo7535889243153246212i_assn,Y: bool,N: nat] :
      ( hoare_1429296392585015714_VEBTi(one_one_assn,X3,aa_boo89690317562068533i_assn(A2,Y))
     => hoare_3904069481286416050_VEBTi(one_one_assn,vEBT_V1859673955506687831_VEBTi(N,X3),vEBT_L1689158095293668553_VEBTi(A2,replicate_bool(N,Y))) ) ).

% repli_emp
tff(fact_3212_repli__emp,axiom,
    ! [X3: heap_T8145700208782473153_VEBTi,A2: fun_VE905575822219544865i_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_1429296392585015714_VEBTi(one_one_assn,X3,aa_VEB3717855479479287634i_assn(A2,Y))
     => hoare_3904069481286416050_VEBTi(one_one_assn,vEBT_V1859673955506687831_VEBTi(N,X3),vEBT_L6296928887356842470_VEBTi(A2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),Y))) ) ).

% repli_emp
tff(fact_3213_repli__emp,axiom,
    ! [X3: heap_Time_Heap_nat,A2: fun_VE4931415397359990460t_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_3067605981109127869le_nat(one_one_assn,X3,aa_VEB8640158415945897581t_assn(A2,Y))
     => hoare_7964568885773372237st_nat(one_one_assn,vEBT_V7726092123322077554ei_nat(N,X3),vEBT_L8296926524756676353BT_nat(A2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),Y))) ) ).

% repli_emp
tff(fact_3214_repli__emp,axiom,
    ! [X3: heap_Time_Heap_nat,A2: fun_bo8438563643002758303t_assn,Y: bool,N: nat] :
      ( hoare_3067605981109127869le_nat(one_one_assn,X3,aa_bool_fun_nat_assn(A2,Y))
     => hoare_7964568885773372237st_nat(one_one_assn,vEBT_V7726092123322077554ei_nat(N,X3),vEBT_L7520539408223738084ol_nat(A2,replicate_bool(N,Y))) ) ).

% repli_emp
tff(fact_3215_repli__cons__repl,axiom,
    ! [Q: assn,X3: heap_T2636463487746394924on_nat,A2: fun_VE6334927249922962060t_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_7629718768684598413on_nat(Q,X3,aa_VEB931861714549014589t_assn(aa_fun1240582285364316732t_assn(aTP_Lamm_hq(Q),A2),Y))
     => hoare_6480275734082232733on_nat(Q,vEBT_V792416675989592002on_nat(N,X3),aa_nat5248546159757461219t_assn(aa_VEB2926451386496479642t_assn(aa_fun7129419364150962969t_assn(aTP_Lamm_hr(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_3216_repli__cons__repl,axiom,
    ! [Q: assn,X3: heap_T2636463487746394924on_nat,A2: fun_bo1795969981296709615t_assn,Y: bool,N: nat] :
      ( hoare_7629718768684598413on_nat(Q,X3,aa_boo2787173234663865760t_assn(aa_fun9080105975142006140t_assn(aTP_Lamm_hs(Q),A2),Y))
     => hoare_6480275734082232733on_nat(Q,vEBT_V792416675989592002on_nat(N,X3),aa_nat5248546159757461219t_assn(aa_boo6521129556348374909t_assn(aa_fun7381918676215250521t_assn(aTP_Lamm_ht(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_3217_repli__cons__repl,axiom,
    ! [Q: assn,X3: heap_Time_Heap_bool,A2: fun_VE1323945509527881313l_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_2774773592644289642e_bool(Q,X3,aa_VEB2879522196611654106l_assn(aa_fun6755658226996116224l_assn(aTP_Lamm_hu(Q),A2),Y))
     => hoare_9026296798553663728t_bool(Q,vEBT_V6044891547498605301i_bool(N,X3),aa_nat2784677144853144906l_assn(aa_VEB3184745993655927913l_assn(aa_fun7303525306808986255l_assn(aTP_Lamm_hv(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_3218_repli__cons__repl,axiom,
    ! [Q: assn,X3: heap_Time_Heap_bool,A2: fun_bo8172996160782635326l_assn,Y: bool,N: nat] :
      ( hoare_2774773592644289642e_bool(Q,X3,aa_boo2938360491098410167l_assn(aa_fun6282432130954631226l_assn(aTP_Lamm_hw(Q),A2),Y))
     => hoare_9026296798553663728t_bool(Q,vEBT_V6044891547498605301i_bool(N,X3),aa_nat2784677144853144906l_assn(aa_boo5646914333422710470l_assn(aa_fun2051762549316634313l_assn(aTP_Lamm_hx(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_3219_repli__cons__repl,axiom,
    ! [Q: assn,X3: heap_T8145700208782473153_VEBTi,A2: fun_bo7535889243153246212i_assn,Y: bool,N: nat] :
      ( hoare_1429296392585015714_VEBTi(Q,X3,aa_boo89690317562068533i_assn(aa_fun3930196489193831100i_assn(aTP_Lamm_hy(Q),A2),Y))
     => hoare_3904069481286416050_VEBTi(Q,vEBT_V1859673955506687831_VEBTi(N,X3),aa_nat6700791848136694008i_assn(aa_boo3536769854965821074i_assn(aa_fun2584265494025193113i_assn(aTP_Lamm_hz(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_3220_repli__cons__repl,axiom,
    ! [Q: assn,X3: heap_T8145700208782473153_VEBTi,A2: fun_VE905575822219544865i_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_1429296392585015714_VEBTi(Q,X3,aa_VEB3717855479479287634i_assn(aa_fun4659777589554175228i_assn(aTP_Lamm_ia(Q),A2),Y))
     => hoare_3904069481286416050_VEBTi(Q,vEBT_V1859673955506687831_VEBTi(N,X3),aa_nat6700791848136694008i_assn(aa_VEB320677594755043119i_assn(aa_fun8635824850342594777i_assn(aTP_Lamm_ib(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_3221_repli__cons__repl,axiom,
    ! [Q: assn,X3: heap_Time_Heap_nat,A2: fun_VE4931415397359990460t_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_3067605981109127869le_nat(Q,X3,aa_VEB8640158415945897581t_assn(aa_fun7634423771582967228t_assn(aTP_Lamm_ic(Q),A2),Y))
     => hoare_7964568885773372237st_nat(Q,vEBT_V7726092123322077554ei_nat(N,X3),aa_nat7967224596217534867t_assn(aa_VEB8402767592425156170t_assn(aa_fun6018615163312975129t_assn(aTP_Lamm_id(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_3222_repli__cons__repl,axiom,
    ! [Q: assn,X3: heap_Time_Heap_nat,A2: fun_bo8438563643002758303t_assn,Y: bool,N: nat] :
      ( hoare_3067605981109127869le_nat(Q,X3,aa_bool_fun_nat_assn(aa_fun8165606209698844796t_assn(aTP_Lamm_ie(Q),A2),Y))
     => hoare_7964568885773372237st_nat(Q,vEBT_V7726092123322077554ei_nat(N,X3),aa_nat7967224596217534867t_assn(aa_boo7576344649022168237t_assn(aa_fun7875716396929150937t_assn(aTP_Lamm_if(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_3223_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,X3) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3 = 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_3224_diff__shunt__var,axiom,
    ! [X3: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X3,Y) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(X3),Y)) ) ).

% diff_shunt_var
tff(fact_3225_diff__shunt__var,axiom,
    ! [X3: set_bool,Y: set_bool] :
      ( ( minus_minus_set_bool(X3,Y) = bot_bot_set_bool )
    <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(X3),Y)) ) ).

% diff_shunt_var
tff(fact_3226_diff__shunt__var,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X3),Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Y)) ) ).

% diff_shunt_var
tff(fact_3227_diff__shunt__var,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,X3),Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y)) ) ).

% diff_shunt_var
tff(fact_3228_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBTi,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3229_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBT,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3230_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_real,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3231_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_bool,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From)))
         => ( pp(aa_nat_bool(nth_bool(slice_bool(From,To,Xs)),I))
          <=> pp(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I))) ) ) ) ) ).

% slice_nth
tff(fact_3232_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3233_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_int_nat(size_size_list_int,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3234_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_3235_intind,axiom,
    ! [I: nat,N: nat,P: fun_nat_bool,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(P,X3))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(replicate_nat(N,X3)),I))) ) ) ).

% intind
tff(fact_3236_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBTi_bool,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBTi_bool(P,X3))
       => pp(aa_VEBT_VEBTi_bool(P,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(replicate_VEBT_VEBTi(N,X3)),I))) ) ) ).

% intind
tff(fact_3237_intind,axiom,
    ! [I: nat,N: nat,P: fun_int_bool,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_int_bool(P,X3))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(replicate_int(N,X3)),I))) ) ) ).

% intind
tff(fact_3238_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBT_bool,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBT_bool(P,X3))
       => pp(aa_VEBT_VEBT_bool(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3)),I))) ) ) ).

% intind
tff(fact_3239_intind,axiom,
    ! [I: nat,N: nat,P: fun_bool_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_bool_bool(P,X3))
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(replicate_bool(N,X3)),I))) ) ) ).

% intind
tff(fact_3240_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_3241_Diff__cancel,axiom,
    ! [A2: set_bool] : minus_minus_set_bool(A2,A2) = bot_bot_set_bool ).

% Diff_cancel
tff(fact_3242_Diff__cancel,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_3243_Diff__cancel,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_3244_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_3245_empty__Diff,axiom,
    ! [A2: set_bool] : minus_minus_set_bool(bot_bot_set_bool,A2) = bot_bot_set_bool ).

% empty_Diff
tff(fact_3246_empty__Diff,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,bot_bot_set_int),A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_3247_empty__Diff,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_3248_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_3249_Diff__empty,axiom,
    ! [A2: set_bool] : minus_minus_set_bool(A2,bot_bot_set_bool) = A2 ).

% Diff_empty
tff(fact_3250_Diff__empty,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_3251_Diff__empty,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_3252_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( minus_minus_set_real(A2,B3) = bot_bot_set_real )
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3)) ) ).

% Diff_eq_empty_iff
tff(fact_3253_Diff__eq__empty__iff,axiom,
    ! [A2: set_bool,B3: set_bool] :
      ( ( minus_minus_set_bool(A2,B3) = bot_bot_set_bool )
    <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),B3)) ) ).

% Diff_eq_empty_iff
tff(fact_3254_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3)) ) ).

% Diff_eq_empty_iff
tff(fact_3255_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B3) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ).

% Diff_eq_empty_iff
tff(fact_3256_replicate__eq__replicate,axiom,
    ! [M: nat,X3: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(M),X3) = aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X3 = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_3257_replicate__eq__replicate,axiom,
    ! [M: nat,X3: bool,N: nat,Y: bool] :
      ( ( replicate_bool(M,X3) = replicate_bool(N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( pp(X3)
          <=> pp(Y) ) ) ) ) ).

% replicate_eq_replicate
tff(fact_3258_length__replicate,axiom,
    ! [N: nat,X3: vEBT_VEBT] : size_s6755466524823107622T_VEBT(aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3)) = N ).

% length_replicate
tff(fact_3259_length__replicate,axiom,
    ! [N: nat,X3: real] : aa_list_real_nat(size_size_list_real,aa_real_list_real(replicate_real(N),X3)) = N ).

% length_replicate
tff(fact_3260_length__replicate,axiom,
    ! [N: nat,X3: bool] : aa_list_bool_nat(size_size_list_bool,replicate_bool(N,X3)) = N ).

% length_replicate
tff(fact_3261_length__replicate,axiom,
    ! [N: nat,X3: nat] : aa_list_nat_nat(size_size_list_nat,replicate_nat(N,X3)) = N ).

% length_replicate
tff(fact_3262_length__replicate,axiom,
    ! [N: nat,X3: int] : aa_list_int_nat(size_size_list_int,replicate_int(N,X3)) = N ).

% length_replicate
tff(fact_3263_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_list_nat_set_nat(set_nat2,replicate_nat(N,A))))
         => pp(aa_nat_bool(P,X4)) )
    <=> ( pp(aa_nat_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_3264_Ball__set__replicate,axiom,
    ! [N: nat,A: real,P: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(replicate_real(N),A))))
         => pp(aa_real_bool(P,X4)) )
    <=> ( pp(aa_real_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_3265_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),A))))
         => pp(aa_VEBT_VEBT_bool(P,X4)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_3266_Ball__set__replicate,axiom,
    ! [N: nat,A: bool,P: fun_bool_bool] :
      ( ! [X4: bool] :
          ( pp(aa_set_bool_bool(member_bool(X4),aa_lis3587183946281582166t_bool(set_bool2,replicate_bool(N,A))))
         => pp(aa_bool_bool(P,X4)) )
    <=> ( pp(aa_bool_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_3267_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_list_nat_set_nat(set_nat2,replicate_nat(N,A))))
          & pp(aa_nat_bool(P,X4)) )
    <=> ( pp(aa_nat_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_3268_Bex__set__replicate,axiom,
    ! [N: nat,A: real,P: fun_real_bool] :
      ( ? [X4: real] :
          ( pp(aa_set_real_bool(member_real(X4),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(replicate_real(N),A))))
          & pp(aa_real_bool(P,X4)) )
    <=> ( pp(aa_real_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_3269_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ? [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),A))))
          & pp(aa_VEBT_VEBT_bool(P,X4)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_3270_Bex__set__replicate,axiom,
    ! [N: nat,A: bool,P: fun_bool_bool] :
      ( ? [X4: bool] :
          ( pp(aa_set_bool_bool(member_bool(X4),aa_lis3587183946281582166t_bool(set_bool2,replicate_bool(N,A))))
          & pp(aa_bool_bool(P,X4)) )
    <=> ( pp(aa_bool_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_3271_in__set__replicate,axiom,
    ! [X3: int,N: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,replicate_int(N,Y))))
    <=> ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3272_in__set__replicate,axiom,
    ! [X3: set_nat,N: nat,Y: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(replicate_set_nat(N,Y))))
    <=> ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3273_in__set__replicate,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,replicate_nat(N,Y))))
    <=> ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3274_in__set__replicate,axiom,
    ! [X3: real,N: nat,Y: real] :
      ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(replicate_real(N),Y))))
    <=> ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3275_in__set__replicate,axiom,
    ! [X3: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),Y))))
    <=> ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3276_in__set__replicate,axiom,
    ! [X3: bool,N: nat,Y: bool] :
      ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,replicate_bool(N,Y))))
    <=> ( ( pp(X3)
        <=> pp(Y) )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_3277_nth__replicate,axiom,
    ! [I: nat,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_nat(nth_nat(replicate_nat(N,X3)),I) = X3 ) ) ).

% nth_replicate
tff(fact_3278_nth__replicate,axiom,
    ! [I: nat,N: nat,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(replicate_VEBT_VEBTi(N,X3)),I) = X3 ) ) ).

% nth_replicate
tff(fact_3279_nth__replicate,axiom,
    ! [I: nat,N: nat,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_int(nth_int(replicate_int(N,X3)),I) = X3 ) ) ).

% nth_replicate
tff(fact_3280_nth__replicate,axiom,
    ! [I: nat,N: nat,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3)),I) = X3 ) ) ).

% nth_replicate
tff(fact_3281_nth__replicate,axiom,
    ! [I: nat,N: nat,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(nth_bool(replicate_bool(N,X3)),I))
      <=> pp(X3) ) ) ).

% nth_replicate
tff(fact_3282_slice__complete,axiom,
    ! [Xs: list_real] : slice_real(zero_zero_nat,aa_list_real_nat(size_size_list_real,Xs),Xs) = Xs ).

% slice_complete
tff(fact_3283_slice__complete,axiom,
    ! [Xs: list_bool] : slice_bool(zero_zero_nat,aa_list_bool_nat(size_size_list_bool,Xs),Xs) = Xs ).

% slice_complete
tff(fact_3284_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_3285_slice__complete,axiom,
    ! [Xs: list_int] : slice_int(zero_zero_nat,aa_list_int_nat(size_size_list_int,Xs),Xs) = Xs ).

% slice_complete
tff(fact_3286_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_real_nat(size_size_list_real,Xs)))
       => ( aa_list_real_nat(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_3287_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( aa_list_bool_nat(size_size_list_bool,slice_bool(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_3288_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3289_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),aa_list_int_nat(size_size_list_int,Xs)))
       => ( aa_list_int_nat(size_size_list_int,slice_int(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_3290_double__diff,axiom,
    ! [A2: set_nat,B3: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),C2))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A2)) = A2 ) ) ) ).

% double_diff
tff(fact_3291_double__diff,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),C2))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C2),A2)) = A2 ) ) ) ).

% double_diff
tff(fact_3292_Diff__subset,axiom,
    ! [A2: set_nat,B3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3)),A2)) ).

% Diff_subset
tff(fact_3293_Diff__subset,axiom,
    ! [A2: set_int,B3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B3)),A2)) ).

% Diff_subset
tff(fact_3294_Diff__mono,axiom,
    ! [A2: set_nat,C2: set_nat,D4: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,D4),B3))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),D4))) ) ) ).

% Diff_mono
tff(fact_3295_Diff__mono,axiom,
    ! [A2: set_int,C2: set_int,D4: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(D4),B3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B3)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C2),D4))) ) ) ).

% Diff_mono
tff(fact_3296_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => ( X = X3 ) )
     => ( aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs)),X3) = Xs ) ) ).

% replicate_length_same
tff(fact_3297_replicate__length__same,axiom,
    ! [Xs: list_real,X3: real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => ( X = X3 ) )
     => ( aa_real_list_real(replicate_real(aa_list_real_nat(size_size_list_real,Xs)),X3) = Xs ) ) ).

% replicate_length_same
tff(fact_3298_replicate__length__same,axiom,
    ! [Xs: list_bool,X3: bool] :
      ( ! [X: bool] :
          ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => ( pp(X)
          <=> pp(X3) ) )
     => ( replicate_bool(aa_list_bool_nat(size_size_list_bool,Xs),X3) = Xs ) ) ).

% replicate_length_same
tff(fact_3299_replicate__length__same,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => ( X = X3 ) )
     => ( replicate_nat(aa_list_nat_nat(size_size_list_nat,Xs),X3) = Xs ) ) ).

% replicate_length_same
tff(fact_3300_replicate__length__same,axiom,
    ! [Xs: list_int,X3: int] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
         => ( X = X3 ) )
     => ( replicate_int(aa_list_int_nat(size_size_list_int,Xs),X3) = Xs ) ) ).

% replicate_length_same
tff(fact_3301_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N: nat,X3: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = N )
     => ( ! [Y3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(Y3),set_set_nat2(Xs)))
           => ( Y3 = X3 ) )
       => ( Xs = replicate_set_nat(N,X3) ) ) ) ).

% replicate_eqI
tff(fact_3302_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N )
     => ( ! [Y3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
           => ( Y3 = X3 ) )
       => ( Xs = aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3) ) ) ) ).

% replicate_eqI
tff(fact_3303_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X3: real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = N )
     => ( ! [Y3: real] :
            ( pp(aa_set_real_bool(member_real(Y3),aa_lis3769085826154824284t_real(set_real2,Xs)))
           => ( Y3 = X3 ) )
       => ( Xs = aa_real_list_real(replicate_real(N),X3) ) ) ) ).

% replicate_eqI
tff(fact_3304_replicate__eqI,axiom,
    ! [Xs: list_bool,N: nat,X3: bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = N )
     => ( ! [Y3: bool] :
            ( pp(aa_set_bool_bool(member_bool(Y3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
           => ( pp(Y3)
            <=> pp(X3) ) )
       => ( Xs = replicate_bool(N,X3) ) ) ) ).

% replicate_eqI
tff(fact_3305_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X3: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = N )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),aa_list_nat_set_nat(set_nat2,Xs)))
           => ( Y3 = X3 ) )
       => ( Xs = replicate_nat(N,X3) ) ) ) ).

% replicate_eqI
tff(fact_3306_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X3: int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = N )
     => ( ! [Y3: int] :
            ( pp(aa_set_int_bool(member_int(Y3),aa_list_int_set_int(set_int2,Xs)))
           => ( Y3 = X3 ) )
       => ( Xs = replicate_int(N,X3) ) ) ) ).

% replicate_eqI
tff(fact_3307_sorted__replicate,axiom,
    ! [N: nat,X3: bool] : pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),replicate_bool(N,X3))) ).

% sorted_replicate
tff(fact_3308_sorted__replicate,axiom,
    ! [N: nat,X3: rat] : sorted_wrt_rat(ord_less_eq_rat,replicate_rat(N,X3)) ).

% sorted_replicate
tff(fact_3309_sorted__replicate,axiom,
    ! [N: nat,X3: num] : sorted_wrt_num(ord_less_eq_num,replicate_num(N,X3)) ).

% sorted_replicate
tff(fact_3310_sorted__replicate,axiom,
    ! [N: nat,X3: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),replicate_nat(N,X3))) ).

% sorted_replicate
tff(fact_3311_sorted__replicate,axiom,
    ! [N: nat,X3: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),replicate_int(N,X3))) ).

% sorted_replicate
tff(fact_3312_subset__minus__empty,axiom,
    ! [A2: set_real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
     => ( minus_minus_set_real(A2,B3) = bot_bot_set_real ) ) ).

% subset_minus_empty
tff(fact_3313_subset__minus__empty,axiom,
    ! [A2: set_bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),B3))
     => ( minus_minus_set_bool(A2,B3) = bot_bot_set_bool ) ) ).

% subset_minus_empty
tff(fact_3314_subset__minus__empty,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_3315_subset__minus__empty,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B3) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_3316_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_nat(aTP_Lamm_ig(K),Lst) = replicate_nat(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_3317_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_real(aTP_Lamm_ih(K),Lst) = aa_real_list_real(replicate_real(size_s6755466524823107622T_VEBT(Lst)),K) ).

% map_replicate_const
tff(fact_3318_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] : map_real_VEBT_VEBT(aTP_Lamm_ii(K),Lst) = aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_real_nat(size_size_list_real,Lst)),K) ).

% map_replicate_const
tff(fact_3319_map__replicate__const,axiom,
    ! [K: bool,Lst: list_real] : map_real_bool(aTP_Lamm_ij(K),Lst) = replicate_bool(aa_list_real_nat(size_size_list_real,Lst),K) ).

% map_replicate_const
tff(fact_3320_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_bool] : map_bool_VEBT_VEBT(aTP_Lamm_ik(K),Lst) = aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_bool_nat(size_size_list_bool,Lst)),K) ).

% map_replicate_const
tff(fact_3321_map__replicate__const,axiom,
    ! [K: bool,Lst: list_bool] : map_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_il,K),Lst) = replicate_bool(aa_list_bool_nat(size_size_list_bool,Lst),K) ).

% map_replicate_const
tff(fact_3322_map__replicate__const,axiom,
    ! [K: nat,Lst: list_nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_im,K),Lst) = replicate_nat(aa_list_nat_nat(size_size_list_nat,Lst),K) ).

% map_replicate_const
tff(fact_3323_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] : map_nat_VEBT_VEBT(aTP_Lamm_in(K),Lst) = aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_nat_nat(size_size_list_nat,Lst)),K) ).

% map_replicate_const
tff(fact_3324_map__replicate__const,axiom,
    ! [K: bool,Lst: list_nat] : map_nat_bool(aa_bool_fun_nat_bool(aTP_Lamm_io,K),Lst) = replicate_bool(aa_list_nat_nat(size_size_list_nat,Lst),K) ).

% map_replicate_const
tff(fact_3325_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_int] : map_int_VEBT_VEBT(aTP_Lamm_ip(K),Lst) = aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_int_nat(size_size_list_int,Lst)),K) ).

% map_replicate_const
tff(fact_3326_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_int,S4: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => finite6197958912794628473et_int(collect_set_int(aa_set7694327796961020398t_bool(aTP_Lamm_iq(S3),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3327_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_complex,S4: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => finite6551019134538273531omplex(collect_set_complex(aa_set6592632205191297002x_bool(aTP_Lamm_ir(S3),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3328_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_Code_integer,S4: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => finite6931041176100689706nteger(collec574505750873337192nteger(aa_set8732061614394825996r_bool(aTP_Lamm_is(S3),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3329_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_nat,S4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_it(S3),S4)))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3330_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = one_one_nat ).

% VEBT_internal.cnt'.simps(1)
tff(fact_3331_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),X3))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = one_one_nat )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = 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)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_3332_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_VEBT_VEBT_set_nat(vEBT_set_vebt,T)),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_3333_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_3334_length__product,axiom,
    ! [Xs: list_real,Ys: list_real] : size_s3932428310213730859l_real(product_real_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_real_nat(size_size_list_real,Ys)) ).

% length_product
tff(fact_3335_length__product,axiom,
    ! [Xs: list_real,Ys: list_bool] : size_s2850477280887376644l_bool(product_real_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_bool_nat(size_size_list_bool,Ys)) ).

% length_product
tff(fact_3336_length__product,axiom,
    ! [Xs: list_real,Ys: list_nat] : size_s1877336372972134351al_nat(product_real_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_product
tff(fact_3337_length__product,axiom,
    ! [Xs: list_real,Ys: list_int] : size_s8610625264895183403al_int(product_real_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_int_nat(size_size_list_int,Ys)) ).

% length_product
tff(fact_3338_length__product,axiom,
    ! [Xs: list_bool,Ys: list_real] : size_s8805918724536545814l_real(product_bool_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_bool_nat(size_size_list_bool,Xs)),aa_list_real_nat(size_size_list_real,Ys)) ).

% length_product
tff(fact_3339_length__product,axiom,
    ! [Xs: list_bool,Ys: list_bool] : size_s6988337448923680857l_bool(product_bool_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_bool_nat(size_size_list_bool,Xs)),aa_list_bool_nat(size_size_list_bool,Ys)) ).

% length_product
tff(fact_3340_length__product,axiom,
    ! [Xs: list_bool,Ys: list_nat] : size_s3243772863075988794ol_nat(product_bool_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_bool_nat(size_size_list_bool,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_product
tff(fact_3341_length__product,axiom,
    ! [Xs: list_bool,Ys: list_int] : size_s753689718144262038ol_int(product_bool_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_bool_nat(size_size_list_bool,Xs)),aa_list_int_nat(size_size_list_int,Ys)) ).

% length_product
tff(fact_3342_length__product,axiom,
    ! [Xs: list_nat,Ys: list_real] : size_s7910714270633306959t_real(product_nat_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_real_nat(size_size_list_real,Ys)) ).

% length_product
tff(fact_3343_length__product,axiom,
    ! [Xs: list_nat,Ys: list_bool] : size_s783543119881238752t_bool(product_nat_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_bool_nat(size_size_list_bool,Ys)) ).

% length_product
tff(fact_3344_product__code,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : produc3886929683002245970T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys)) = set_Pr9182192707038809660T_VEBT(concat8823785421885603304T_VEBT(map_VE6710161768887062461T_VEBT(aTP_Lamm_iu(Ys),Xs))) ).

% product_code
tff(fact_3345_product__code,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real] : produc8336039005996768526T_real(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_lis3769085826154824284t_real(set_real2,Ys)) = set_Pr1087130671499945274T_real(concat8867029399158321294T_real(map_VE6455195641817765263T_real(aTP_Lamm_iv(Ys),Xs))) ).

% product_code
tff(fact_3346_product__code,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool] : produc5189178589784285493T_bool(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_lis3587183946281582166t_bool(set_bool2,Ys)) = set_Pr3399341499479959199T_bool(concat7195019909724358219T_bool(map_VE8357768277379645088T_bool(aTP_Lamm_iw(Ys),Xs))) ).

% product_code
tff(fact_3347_product__code,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : produc1441754205559175824T_VEBT(aa_list_nat_set_nat(set_nat2,Xs),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys)) = set_Pr5984661752051438084T_VEBT(concat3360658388503054168T_VEBT(map_na6744855865280072703T_VEBT(aTP_Lamm_ix(Ys),Xs))) ).

% product_code
tff(fact_3348_product__code,axiom,
    ! [Xs: list_nat,Ys: list_real] : produc6734486746367016272t_real(aa_list_nat_set_nat(set_nat2,Xs),aa_lis3769085826154824284t_real(set_real2,Ys)) = set_Pr7149346036329476978t_real(concat8049025717632320286t_real(map_na1385415807590506957t_real(aTP_Lamm_iy(Ys),Xs))) ).

% product_code
tff(fact_3349_product__code,axiom,
    ! [Xs: list_nat,Ys: list_bool] : produc3346841887225286771t_bool(aa_list_nat_set_nat(set_nat2,Xs),aa_lis3587183946281582166t_bool(set_bool2,Ys)) = set_Pr3163503208309705319t_bool(concat2246367945322123195t_bool(map_na6080073900867125986t_bool(aTP_Lamm_iz(Ys),Xs))) ).

% product_code
tff(fact_3350_product__code,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT] : produc7150050738623674420T_VEBT(aa_lis3769085826154824284t_real(set_real2,Xs),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys)) = set_Pr8897343066327330088T_VEBT(concat7453869757130930300T_VEBT(map_re4732419559776791935T_VEBT(aTP_Lamm_ja(Ys),Xs))) ).

% product_code
tff(fact_3351_product__code,axiom,
    ! [Xs: list_real,Ys: list_nat] : produc2078423282641139152al_nat(aa_lis3769085826154824284t_real(set_real2,Xs),aa_list_nat_set_nat(set_nat2,Ys)) = set_Pr3174298344852596722al_nat(concat4073978026155440030al_nat(map_re2667248788615319281al_nat(aTP_Lamm_jb(Ys),Xs))) ).

% product_code
tff(fact_3352_product__code,axiom,
    ! [Xs: list_real,Ys: list_real] : produc2998788613821421740l_real(aa_lis3769085826154824284t_real(set_real2,Xs),aa_lis3769085826154824284t_real(set_real2,Ys)) = set_Pr5999470521830281550l_real(concat431446536988871418l_real(map_re4578740227252052045l_real(aTP_Lamm_jc(Ys),Xs))) ).

% product_code
tff(fact_3353_product__code,axiom,
    ! [Xs: list_real,Ys: list_bool] : produc9024588101255028247l_bool(aa_lis3769085826154824284t_real(set_real2,Xs),aa_lis3587183946281582166t_bool(set_bool2,Ys)) = set_Pr3131006541957657995l_bool(concat860756493465424095l_bool(map_re3410928008045110370l_bool(aTP_Lamm_jd(Ys),Xs))) ).

% product_code
tff(fact_3354_prod__decode__aux_Oelims,axiom,
    ! [X3: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X3,Xa) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X3))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Xa)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X3))
         => ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X3))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_3355_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( nat_prod_decode_aux(K,M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( nat_prod_decode_aux(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_3356_diff__diff__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N)) ) ) ).

% diff_diff_less
tff(fact_3357_singletonI,axiom,
    ! [A: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))) ).

% singletonI
tff(fact_3358_singletonI,axiom,
    ! [A: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(A),aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat))) ).

% singletonI
tff(fact_3359_singletonI,axiom,
    ! [A: real] : pp(aa_set_real_bool(member_real(A),aa_set_real_set_real(insert_real(A),bot_bot_set_real))) ).

% singletonI
tff(fact_3360_singletonI,axiom,
    ! [A: bool] : pp(aa_set_bool_bool(member_bool(A),aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool))) ).

% singletonI
tff(fact_3361_singletonI,axiom,
    ! [A: nat] : pp(aa_set_nat_bool(member_nat(A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) ).

% singletonI
tff(fact_3362_singletonI,axiom,
    ! [A: int] : pp(aa_set_int_bool(member_int(A),aa_set_int_set_int(insert_int(A),bot_bot_set_int))) ).

% singletonI
tff(fact_3363_insert__subset,axiom,
    ! [X3: bool,A2: set_bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_set_bool_set_bool(insert_bool(X3),A2)),B3))
    <=> ( pp(aa_set_bool_bool(member_bool(X3),B3))
        & pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),B3)) ) ) ).

% insert_subset
tff(fact_3364_insert__subset,axiom,
    ! [X3: nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(insert_nat(X3),A2)),B3))
    <=> ( pp(aa_set_nat_bool(member_nat(X3),B3))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3)) ) ) ).

% insert_subset
tff(fact_3365_insert__subset,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),A2)),B3))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),B3))
        & pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3)) ) ) ).

% insert_subset
tff(fact_3366_insert__subset,axiom,
    ! [X3: real,A2: set_real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_set_real_set_real(insert_real(X3),A2)),B3))
    <=> ( pp(aa_set_real_bool(member_real(X3),B3))
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3)) ) ) ).

% insert_subset
tff(fact_3367_insert__subset,axiom,
    ! [X3: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(aa_set1013257548857649334et_nat(insert_set_nat(X3),A2)),B3))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(X3),B3))
        & pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3)) ) ) ).

% insert_subset
tff(fact_3368_insert__subset,axiom,
    ! [X3: int,A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(insert_int(X3),A2)),B3))
    <=> ( pp(aa_set_int_bool(member_int(X3),B3))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ) ).

% insert_subset
tff(fact_3369_singleton__conv,axiom,
    ! [A: vEBT_VEBT] : collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_je,A)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ).

% singleton_conv
tff(fact_3370_singleton__conv,axiom,
    ! [A: complex] : collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_jf,A)) = insert_complex(A,bot_bot_set_complex) ).

% singleton_conv
tff(fact_3371_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jg,A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_3372_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jh,A)) = aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_3373_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ji,A)) = aa_set_real_set_real(insert_real(A),bot_bot_set_real) ).

% singleton_conv
tff(fact_3374_singleton__conv,axiom,
    ! [A: bool] : collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_jj,A)) = aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool) ).

% singleton_conv
tff(fact_3375_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jk,A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ).

% singleton_conv
tff(fact_3376_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_bool(aTP_Lamm_jl,A)) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ).

% singleton_conv
tff(fact_3377_singleton__conv2,axiom,
    ! [A: vEBT_VEBT] : collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,A)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ).

% singleton_conv2
tff(fact_3378_singleton__conv2,axiom,
    ! [A: complex] : collect_complex(aa_com1553036987509291606x_bool(fequal_complex,A)) = insert_complex(A,bot_bot_set_complex) ).

% singleton_conv2
tff(fact_3379_singleton__conv2,axiom,
    ! [A: list_nat] : collect_list_nat(aa_lis4768068104879709938t_bool(fequal_list_nat,A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv2
tff(fact_3380_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(aa_set5116768664143533222t_bool(fequal_set_nat,A)) = aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_3381_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_rea7809428808904475482l_bool(fequal_real,A)) = aa_set_real_set_real(insert_real(A),bot_bot_set_real) ).

% singleton_conv2
tff(fact_3382_singleton__conv2,axiom,
    ! [A: bool] : collect_bool(aa_boo8970175072276269406l_bool(fequal_bool,A)) = aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool) ).

% singleton_conv2
tff(fact_3383_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_bool(fequal_nat,A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ).

% singleton_conv2
tff(fact_3384_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_bool(fequal_int,A)) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ).

% singleton_conv2
tff(fact_3385_singleton__insert__inj__eq_H,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A2) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT) )
    <=> ( ( A = B )
        & pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_3386_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( aa_set_real_set_real(insert_real(A),A2) = aa_set_real_set_real(insert_real(B),bot_bot_set_real) )
    <=> ( ( A = B )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),aa_set_real_set_real(insert_real(B),bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_3387_singleton__insert__inj__eq_H,axiom,
    ! [A: bool,A2: set_bool,B: bool] :
      ( ( aa_set_bool_set_bool(insert_bool(A),A2) = aa_set_bool_set_bool(insert_bool(B),bot_bot_set_bool) )
    <=> ( ( pp(A)
        <=> pp(B) )
        & pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),aa_set_bool_set_bool(insert_bool(B),bot_bot_set_bool))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_3388_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( aa_set_nat_set_nat(insert_nat(A),A2) = aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) )
    <=> ( ( A = B )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_3389_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( aa_set_int_set_int(insert_int(A),A2) = aa_set_int_set_int(insert_int(B),bot_bot_set_int) )
    <=> ( ( A = B )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(insert_int(B),bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_3390_singleton__insert__inj__eq,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A2) )
    <=> ( ( A = B )
        & pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT))) ) ) ).

% singleton_insert_inj_eq
tff(fact_3391_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( aa_set_real_set_real(insert_real(B),bot_bot_set_real) = aa_set_real_set_real(insert_real(A),A2) )
    <=> ( ( A = B )
        & pp(aa_set_real_bool(ord_less_eq_set_real(A2),aa_set_real_set_real(insert_real(B),bot_bot_set_real))) ) ) ).

% singleton_insert_inj_eq
tff(fact_3392_singleton__insert__inj__eq,axiom,
    ! [B: bool,A: bool,A2: set_bool] :
      ( ( aa_set_bool_set_bool(insert_bool(B),bot_bot_set_bool) = aa_set_bool_set_bool(insert_bool(A),A2) )
    <=> ( ( pp(A)
        <=> pp(B) )
        & pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),aa_set_bool_set_bool(insert_bool(B),bot_bot_set_bool))) ) ) ).

% singleton_insert_inj_eq
tff(fact_3393_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) = aa_set_nat_set_nat(insert_nat(A),A2) )
    <=> ( ( A = B )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_3394_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( aa_set_int_set_int(insert_int(B),bot_bot_set_int) = aa_set_int_set_int(insert_int(A),A2) )
    <=> ( ( A = B )
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(insert_int(B),bot_bot_set_int))) ) ) ).

% singleton_insert_inj_eq
tff(fact_3395_insert__Diff__single,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] : aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A2) ).

% insert_Diff_single
tff(fact_3396_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : aa_set_real_set_real(insert_real(A),minus_minus_set_real(A2,aa_set_real_set_real(insert_real(A),bot_bot_set_real))) = aa_set_real_set_real(insert_real(A),A2) ).

% insert_Diff_single
tff(fact_3397_insert__Diff__single,axiom,
    ! [A: bool,A2: set_bool] : aa_set_bool_set_bool(insert_bool(A),minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool))) = aa_set_bool_set_bool(insert_bool(A),A2) ).

% insert_Diff_single
tff(fact_3398_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : aa_set_int_set_int(insert_int(A),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(A),bot_bot_set_int))) = aa_set_int_set_int(insert_int(A),A2) ).

% insert_Diff_single
tff(fact_3399_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = aa_set_nat_set_nat(insert_nat(A),A2) ).

% insert_Diff_single
tff(fact_3400_set__replicate,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_3401_set__replicate,axiom,
    ! [N: nat,X3: real] :
      ( ( N != zero_zero_nat )
     => ( aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(replicate_real(N),X3)) = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_3402_set__replicate,axiom,
    ! [N: nat,X3: bool] :
      ( ( N != zero_zero_nat )
     => ( aa_lis3587183946281582166t_bool(set_bool2,replicate_bool(N,X3)) = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) ) ) ).

% set_replicate
tff(fact_3403_set__replicate,axiom,
    ! [N: nat,X3: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_list_nat_set_nat(set_nat2,replicate_nat(N,X3)) = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_3404_set__replicate,axiom,
    ! [N: nat,X3: int] :
      ( ( N != zero_zero_nat )
     => ( aa_list_int_set_int(set_int2,replicate_int(N,X3)) = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_3405_set__diff__eq,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B3) = collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_jm(A2),B3)) ).

% set_diff_eq
tff(fact_3406_set__diff__eq,axiom,
    ! [A2: set_real,B3: set_real] : minus_minus_set_real(A2,B3) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_jn(A2),B3)) ).

% set_diff_eq
tff(fact_3407_set__diff__eq,axiom,
    ! [A2: set_complex,B3: set_complex] : minus_811609699411566653omplex(A2,B3) = collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_jo(A2),B3)) ).

% set_diff_eq
tff(fact_3408_set__diff__eq,axiom,
    ! [A2: set_list_nat,B3: set_list_nat] : minus_7954133019191499631st_nat(A2,B3) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_jp(A2),B3)) ).

% set_diff_eq
tff(fact_3409_set__diff__eq,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A2,B3) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_jq(A2),B3)) ).

% set_diff_eq
tff(fact_3410_set__diff__eq,axiom,
    ! [A2: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B3) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_jr(A2),B3)) ).

% set_diff_eq
tff(fact_3411_set__diff__eq,axiom,
    ! [A2: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_js(A2),B3)) ).

% set_diff_eq
tff(fact_3412_minus__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,B3) = collect_VEBT_VEBT(minus_1873103217924207103T_bool(aa_set1728193350978918526T_bool(aTP_Lamm_an,A2),aa_set1728193350978918526T_bool(aTP_Lamm_an,B3))) ).

% minus_set_def
tff(fact_3413_minus__set__def,axiom,
    ! [A2: set_real,B3: set_real] : minus_minus_set_real(A2,B3) = collect_real(minus_6582619742941512235l_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ao,A2),aa_set8565328311858951076l_bool(aTP_Lamm_ao,B3))) ).

% minus_set_def
tff(fact_3414_minus__set__def,axiom,
    ! [A2: set_complex,B3: set_complex] : minus_811609699411566653omplex(A2,B3) = collect_complex(minus_5423061390740260265x_bool(aTP_Lamm_ap(A2),aTP_Lamm_ap(B3))) ).

% minus_set_def
tff(fact_3415_minus__set__def,axiom,
    ! [A2: set_list_nat,B3: set_list_nat] : minus_7954133019191499631st_nat(A2,B3) = collect_list_nat(minus_8064998606961450743t_bool(aTP_Lamm_aq(A2),aTP_Lamm_aq(B3))) ).

% minus_set_def
tff(fact_3416_minus__set__def,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A2,B3) = collect_set_nat(minus_8322047804725131473t_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ar,A2),aa_set9121396911425987952t_bool(aTP_Lamm_ar,B3))) ).

% minus_set_def
tff(fact_3417_minus__set__def,axiom,
    ! [A2: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B3) = collect_int(minus_7727386867957376939t_bool(aa_set7012339369581096484t_bool(aTP_Lamm_at,A2),aa_set7012339369581096484t_bool(aTP_Lamm_at,B3))) ).

% minus_set_def
tff(fact_3418_minus__set__def,axiom,
    ! [A2: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3) = collect_nat(minus_1628305797571225351t_bool(aa_set3744802623701456604t_bool(aTP_Lamm_as,A2),aa_set3744802623701456604t_bool(aTP_Lamm_as,B3))) ).

% minus_set_def
tff(fact_3419_insert__Collect,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] : aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),collect_VEBT_VEBT(P)) = collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_jt(A),P)) ).

% insert_Collect
tff(fact_3420_insert__Collect,axiom,
    ! [A: bool,P: fun_bool_bool] : aa_set_bool_set_bool(insert_bool(A),collect_bool(P)) = collect_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_ju(A),P)) ).

% insert_Collect
tff(fact_3421_insert__Collect,axiom,
    ! [A: real,P: fun_real_bool] : aa_set_real_set_real(insert_real(A),collect_real(P)) = collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_jv(A),P)) ).

% insert_Collect
tff(fact_3422_insert__Collect,axiom,
    ! [A: complex,P: fun_complex_bool] : insert_complex(A,collect_complex(P)) = collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_jw(A),P)) ).

% insert_Collect
tff(fact_3423_insert__Collect,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] : insert_list_nat(A,collect_list_nat(P)) = collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_jx(A),P)) ).

% insert_Collect
tff(fact_3424_insert__Collect,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] : aa_set1013257548857649334et_nat(insert_set_nat(A),collect_set_nat(P)) = collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_jy(A),P)) ).

% insert_Collect
tff(fact_3425_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_bool] : aa_set_nat_set_nat(insert_nat(A),collect_nat(P)) = collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_jz(A),P)) ).

% insert_Collect
tff(fact_3426_insert__Collect,axiom,
    ! [A: int,P: fun_int_bool] : aa_set_int_set_int(insert_int(A),collect_int(P)) = collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_ka(A),P)) ).

% insert_Collect
tff(fact_3427_insert__compr,axiom,
    ! [A: bool,B3: set_bool] : aa_set_bool_set_bool(insert_bool(A),B3) = collect_bool(aa_set6670506438128111678l_bool(aTP_Lamm_kb(A),B3)) ).

% insert_compr
tff(fact_3428_insert__compr,axiom,
    ! [A: vEBT_VEBT,B3: set_VEBT_VEBT] : aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B3) = collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_kc(A),B3)) ).

% insert_compr
tff(fact_3429_insert__compr,axiom,
    ! [A: real,B3: set_real] : aa_set_real_set_real(insert_real(A),B3) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_kd(A),B3)) ).

% insert_compr
tff(fact_3430_insert__compr,axiom,
    ! [A: complex,B3: set_complex] : insert_complex(A,B3) = collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_ke(A),B3)) ).

% insert_compr
tff(fact_3431_insert__compr,axiom,
    ! [A: list_nat,B3: set_list_nat] : insert_list_nat(A,B3) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_kf(A),B3)) ).

% insert_compr
tff(fact_3432_insert__compr,axiom,
    ! [A: set_nat,B3: set_set_nat] : aa_set1013257548857649334et_nat(insert_set_nat(A),B3) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_kg(A),B3)) ).

% insert_compr
tff(fact_3433_insert__compr,axiom,
    ! [A: nat,B3: set_nat] : aa_set_nat_set_nat(insert_nat(A),B3) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_kh(A),B3)) ).

% insert_compr
tff(fact_3434_insert__compr,axiom,
    ! [A: int,B3: set_int] : aa_set_int_set_int(insert_int(A),B3) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_ki(A),B3)) ).

% insert_compr
tff(fact_3435_singletonD,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))
     => ( B = A ) ) ).

% singletonD
tff(fact_3436_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B),aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_3437_singletonD,axiom,
    ! [B: real,A: real] :
      ( pp(aa_set_real_bool(member_real(B),aa_set_real_set_real(insert_real(A),bot_bot_set_real)))
     => ( B = A ) ) ).

% singletonD
tff(fact_3438_singletonD,axiom,
    ! [B: bool,A: bool] :
      ( pp(aa_set_bool_bool(member_bool(B),aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool)))
     => ( pp(B)
      <=> pp(A) ) ) ).

% singletonD
tff(fact_3439_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_set_nat_bool(member_nat(B),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))
     => ( B = A ) ) ).

% singletonD
tff(fact_3440_singletonD,axiom,
    ! [B: int,A: int] :
      ( pp(aa_set_int_bool(member_int(B),aa_set_int_set_int(insert_int(A),bot_bot_set_int)))
     => ( B = A ) ) ).

% singletonD
tff(fact_3441_singleton__iff,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_3442_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B),aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_3443_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_set_real_bool(member_real(B),aa_set_real_set_real(insert_real(A),bot_bot_set_real)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_3444_singleton__iff,axiom,
    ! [B: bool,A: bool] :
      ( pp(aa_set_bool_bool(member_bool(B),aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool)))
    <=> ( pp(B)
      <=> pp(A) ) ) ).

% singleton_iff
tff(fact_3445_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_set_nat_bool(member_nat(B),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_3446_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_set_int_bool(member_int(B),aa_set_int_set_int(insert_int(A),bot_bot_set_int)))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_3447_doubleton__eq__iff,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,C: 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(C),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(D),bot_bo8194388402131092736T_VEBT)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_3448_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_set_real_set_real(insert_real(A),aa_set_real_set_real(insert_real(B),bot_bot_set_real)) = aa_set_real_set_real(insert_real(C),aa_set_real_set_real(insert_real(D),bot_bot_set_real)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_3449_doubleton__eq__iff,axiom,
    ! [A: bool,B: bool,C: bool,D: bool] :
      ( ( aa_set_bool_set_bool(insert_bool(A),aa_set_bool_set_bool(insert_bool(B),bot_bot_set_bool)) = aa_set_bool_set_bool(insert_bool(C),aa_set_bool_set_bool(insert_bool(D),bot_bot_set_bool)) )
    <=> ( ( ( pp(A)
          <=> pp(C) )
          & ( pp(B)
          <=> pp(D) ) )
        | ( ( pp(A)
          <=> pp(D) )
          & ( pp(B)
          <=> pp(C) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_3450_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: 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(C),aa_set_nat_set_nat(insert_nat(D),bot_bot_set_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_3451_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: 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(C),aa_set_int_set_int(insert_int(D),bot_bot_set_int)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_3452_insert__not__empty,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] : aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A2) != bot_bo8194388402131092736T_VEBT ).

% insert_not_empty
tff(fact_3453_insert__not__empty,axiom,
    ! [A: real,A2: set_real] : aa_set_real_set_real(insert_real(A),A2) != bot_bot_set_real ).

% insert_not_empty
tff(fact_3454_insert__not__empty,axiom,
    ! [A: bool,A2: set_bool] : aa_set_bool_set_bool(insert_bool(A),A2) != bot_bot_set_bool ).

% insert_not_empty
tff(fact_3455_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : aa_set_nat_set_nat(insert_nat(A),A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_3456_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : aa_set_int_set_int(insert_int(A),A2) != bot_bot_set_int ).

% insert_not_empty
tff(fact_3457_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_3458_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( aa_set_real_set_real(insert_real(A),bot_bot_set_real) = aa_set_real_set_real(insert_real(B),bot_bot_set_real) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_3459_singleton__inject,axiom,
    ! [A: bool,B: bool] :
      ( ( aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool) = aa_set_bool_set_bool(insert_bool(B),bot_bot_set_bool) )
     => ( pp(A)
      <=> pp(B) ) ) ).

% singleton_inject
tff(fact_3460_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_3461_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_3462_set__minus__singleton__eq,axiom,
    ! [X3: vEBT_VEBT,X6: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),X6))
     => ( minus_5127226145743854075T_VEBT(X6,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_3463_set__minus__singleton__eq,axiom,
    ! [X3: set_nat,X6: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),X6))
     => ( minus_2163939370556025621et_nat(X6,aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_3464_set__minus__singleton__eq,axiom,
    ! [X3: real,X6: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X3),X6))
     => ( minus_minus_set_real(X6,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_3465_set__minus__singleton__eq,axiom,
    ! [X3: bool,X6: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(X3),X6))
     => ( minus_minus_set_bool(X6,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_3466_set__minus__singleton__eq,axiom,
    ! [X3: int,X6: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X3),X6))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,X6),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_3467_set__minus__singleton__eq,axiom,
    ! [X3: nat,X6: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X3),X6))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X6),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_3468_Diff__insert__absorb,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
     => ( minus_5127226145743854075T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_3469_Diff__insert__absorb,axiom,
    ! [X3: set_nat,A2: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
     => ( minus_2163939370556025621et_nat(aa_set1013257548857649334et_nat(insert_set_nat(X3),A2),aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_3470_Diff__insert__absorb,axiom,
    ! [X3: real,A2: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X3),A2))
     => ( minus_minus_set_real(aa_set_real_set_real(insert_real(X3),A2),aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_3471_Diff__insert__absorb,axiom,
    ! [X3: bool,A2: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(X3),A2))
     => ( minus_minus_set_bool(aa_set_bool_set_bool(insert_bool(X3),A2),aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_3472_Diff__insert__absorb,axiom,
    ! [X3: int,A2: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X3),A2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(insert_int(X3),A2)),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_3473_Diff__insert__absorb,axiom,
    ! [X3: nat,A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X3),A2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(insert_nat(X3),A2)),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_3474_insert__minus__eq,axiom,
    ! [X3: vEBT_VEBT,Y: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( X3 != Y )
     => ( minus_5127226145743854075T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Y),bot_bo8194388402131092736T_VEBT)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Y),bot_bo8194388402131092736T_VEBT))) ) ) ).

% insert_minus_eq
tff(fact_3475_insert__minus__eq,axiom,
    ! [X3: real,Y: real,A2: set_real] :
      ( ( X3 != Y )
     => ( minus_minus_set_real(aa_set_real_set_real(insert_real(X3),A2),aa_set_real_set_real(insert_real(Y),bot_bot_set_real)) = aa_set_real_set_real(insert_real(X3),minus_minus_set_real(A2,aa_set_real_set_real(insert_real(Y),bot_bot_set_real))) ) ) ).

% insert_minus_eq
tff(fact_3476_insert__minus__eq,axiom,
    ! [X3: bool,Y: bool,A2: set_bool] :
      ( ~ ( pp(X3)
        <=> pp(Y) )
     => ( minus_minus_set_bool(aa_set_bool_set_bool(insert_bool(X3),A2),aa_set_bool_set_bool(insert_bool(Y),bot_bot_set_bool)) = aa_set_bool_set_bool(insert_bool(X3),minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(Y),bot_bot_set_bool))) ) ) ).

% insert_minus_eq
tff(fact_3477_insert__minus__eq,axiom,
    ! [X3: int,Y: int,A2: set_int] :
      ( ( X3 != Y )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(insert_int(X3),A2)),aa_set_int_set_int(insert_int(Y),bot_bot_set_int)) = aa_set_int_set_int(insert_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(Y),bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_3478_insert__minus__eq,axiom,
    ! [X3: nat,Y: nat,A2: set_nat] :
      ( ( X3 != Y )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(insert_nat(X3),A2)),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)) = aa_set_nat_set_nat(insert_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_3479_Diff__insert2,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B3: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B3)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)),B3) ).

% Diff_insert2
tff(fact_3480_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B3: set_real] : minus_minus_set_real(A2,aa_set_real_set_real(insert_real(A),B3)) = minus_minus_set_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(A),bot_bot_set_real)),B3) ).

% Diff_insert2
tff(fact_3481_Diff__insert2,axiom,
    ! [A2: set_bool,A: bool,B3: set_bool] : minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(A),B3)) = minus_minus_set_bool(minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool)),B3) ).

% Diff_insert2
tff(fact_3482_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(A),B3)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(A),bot_bot_set_int))),B3) ).

% Diff_insert2
tff(fact_3483_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))),B3) ).

% Diff_insert2
tff(fact_3484_insert__Diff,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),A2))
     => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))) = A2 ) ) ).

% insert_Diff
tff(fact_3485_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( aa_set1013257548857649334et_nat(insert_set_nat(A),minus_2163939370556025621et_nat(A2,aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_3486_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( aa_set_real_set_real(insert_real(A),minus_minus_set_real(A2,aa_set_real_set_real(insert_real(A),bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_3487_insert__Diff,axiom,
    ! [A: bool,A2: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(A),A2))
     => ( aa_set_bool_set_bool(insert_bool(A),minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool))) = A2 ) ) ).

% insert_Diff
tff(fact_3488_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ( aa_set_int_set_int(insert_int(A),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(A),bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_3489_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_3490_Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B3: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B3)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A2,B3),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)) ).

% Diff_insert
tff(fact_3491_Diff__insert,axiom,
    ! [A2: set_real,A: real,B3: set_real] : minus_minus_set_real(A2,aa_set_real_set_real(insert_real(A),B3)) = minus_minus_set_real(minus_minus_set_real(A2,B3),aa_set_real_set_real(insert_real(A),bot_bot_set_real)) ).

% Diff_insert
tff(fact_3492_Diff__insert,axiom,
    ! [A2: set_bool,A: bool,B3: set_bool] : minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(A),B3)) = minus_minus_set_bool(minus_minus_set_bool(A2,B3),aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool)) ).

% Diff_insert
tff(fact_3493_Diff__insert,axiom,
    ! [A2: set_int,A: int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(A),B3)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B3)),aa_set_int_set_int(insert_int(A),bot_bot_set_int)) ).

% Diff_insert
tff(fact_3494_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(A),B3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3)),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)) ).

% Diff_insert
tff(fact_3495_insert__subsetI,axiom,
    ! [X3: bool,A2: set_bool,X6: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(X3),A2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(X6),A2))
       => pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_set_bool_set_bool(insert_bool(X3),X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_3496_insert__subsetI,axiom,
    ! [X3: nat,A2: set_nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(insert_nat(X3),X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_3497_insert__subsetI,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(X6),A2))
       => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_3498_insert__subsetI,axiom,
    ! [X3: real,A2: set_real,X6: set_real] :
      ( pp(aa_set_real_bool(member_real(X3),A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),A2))
       => pp(aa_set_real_bool(ord_less_eq_set_real(aa_set_real_set_real(insert_real(X3),X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_3499_insert__subsetI,axiom,
    ! [X3: set_nat,A2: set_set_nat,X6: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(X6),A2))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(aa_set1013257548857649334et_nat(insert_set_nat(X3),X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_3500_insert__subsetI,axiom,
    ! [X3: int,A2: set_int,X6: set_int] :
      ( pp(aa_set_int_bool(member_int(X3),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(insert_int(X3),X6)),A2)) ) ) ).

% insert_subsetI
tff(fact_3501_insert__mono,axiom,
    ! [C2: set_nat,D4: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),D4))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(insert_nat(A),C2)),aa_set_nat_set_nat(insert_nat(A),D4))) ) ).

% insert_mono
tff(fact_3502_insert__mono,axiom,
    ! [C2: set_VEBT_VEBT,D4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(C2),D4))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),C2)),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),D4))) ) ).

% insert_mono
tff(fact_3503_insert__mono,axiom,
    ! [C2: set_bool,D4: set_bool,A: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(C2),D4))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_set_bool_set_bool(insert_bool(A),C2)),aa_set_bool_set_bool(insert_bool(A),D4))) ) ).

% insert_mono
tff(fact_3504_insert__mono,axiom,
    ! [C2: set_real,D4: set_real,A: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(C2),D4))
     => pp(aa_set_real_bool(ord_less_eq_set_real(aa_set_real_set_real(insert_real(A),C2)),aa_set_real_set_real(insert_real(A),D4))) ) ).

% insert_mono
tff(fact_3505_insert__mono,axiom,
    ! [C2: set_int,D4: set_int,A: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),D4))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(insert_int(A),C2)),aa_set_int_set_int(insert_int(A),D4))) ) ).

% insert_mono
tff(fact_3506_subset__insert,axiom,
    ! [X3: bool,A2: set_bool,B3: set_bool] :
      ( ~ pp(aa_set_bool_bool(member_bool(X3),A2))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),aa_set_bool_set_bool(insert_bool(X3),B3)))
      <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),B3)) ) ) ).

% subset_insert
tff(fact_3507_subset__insert,axiom,
    ! [X3: nat,A2: set_nat,B3: set_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X3),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),B3)))
      <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3)) ) ) ).

% subset_insert
tff(fact_3508_subset__insert,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),B3)))
      <=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3)) ) ) ).

% subset_insert
tff(fact_3509_subset__insert,axiom,
    ! [X3: real,A2: set_real,B3: set_real] :
      ( ~ pp(aa_set_real_bool(member_real(X3),A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),aa_set_real_set_real(insert_real(X3),B3)))
      <=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3)) ) ) ).

% subset_insert
tff(fact_3510_subset__insert,axiom,
    ! [X3: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),aa_set1013257548857649334et_nat(insert_set_nat(X3),B3)))
      <=> pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3)) ) ) ).

% subset_insert
tff(fact_3511_subset__insert,axiom,
    ! [X3: int,A2: set_int,B3: set_int] :
      ( ~ pp(aa_set_int_bool(member_int(X3),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(insert_int(X3),B3)))
      <=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ) ).

% subset_insert
tff(fact_3512_subset__insertI,axiom,
    ! [B3: set_nat,A: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),aa_set_nat_set_nat(insert_nat(A),B3))) ).

% subset_insertI
tff(fact_3513_subset__insertI,axiom,
    ! [B3: set_VEBT_VEBT,A: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B3),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B3))) ).

% subset_insertI
tff(fact_3514_subset__insertI,axiom,
    ! [B3: set_bool,A: bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(B3),aa_set_bool_set_bool(insert_bool(A),B3))) ).

% subset_insertI
tff(fact_3515_subset__insertI,axiom,
    ! [B3: set_real,A: real] : pp(aa_set_real_bool(ord_less_eq_set_real(B3),aa_set_real_set_real(insert_real(A),B3))) ).

% subset_insertI
tff(fact_3516_subset__insertI,axiom,
    ! [B3: set_int,A: int] : pp(aa_set_int_bool(ord_less_eq_set_int(B3),aa_set_int_set_int(insert_int(A),B3))) ).

% subset_insertI
tff(fact_3517_subset__insertI2,axiom,
    ! [A2: set_nat,B3: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(B),B3))) ) ).

% subset_insertI2
tff(fact_3518_subset__insertI2,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),B3))) ) ).

% subset_insertI2
tff(fact_3519_subset__insertI2,axiom,
    ! [A2: set_bool,B3: set_bool,B: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),B3))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),aa_set_bool_set_bool(insert_bool(B),B3))) ) ).

% subset_insertI2
tff(fact_3520_subset__insertI2,axiom,
    ! [A2: set_real,B3: set_real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),aa_set_real_set_real(insert_real(B),B3))) ) ).

% subset_insertI2
tff(fact_3521_subset__insertI2,axiom,
    ! [A2: set_int,B3: set_int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(insert_int(B),B3))) ) ).

% subset_insertI2
tff(fact_3522_subset__Diff__insert,axiom,
    ! [A2: set_bool,B3: set_bool,X3: bool,C2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),minus_minus_set_bool(B3,aa_set_bool_set_bool(insert_bool(X3),C2))))
    <=> ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),minus_minus_set_bool(B3,C2)))
        & ~ pp(aa_set_bool_bool(member_bool(X3),A2)) ) ) ).

% subset_Diff_insert
tff(fact_3523_subset__Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,X3: vEBT_VEBT,C2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),C2))))
    <=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B3,C2)))
        & ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2)) ) ) ).

% subset_Diff_insert
tff(fact_3524_subset__Diff__insert,axiom,
    ! [A2: set_real,B3: set_real,X3: real,C2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B3,aa_set_real_set_real(insert_real(X3),C2))))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B3,C2)))
        & ~ pp(aa_set_real_bool(member_real(X3),A2)) ) ) ).

% subset_Diff_insert
tff(fact_3525_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,X3: set_nat,C2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B3,aa_set1013257548857649334et_nat(insert_set_nat(X3),C2))))
    <=> ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B3,C2)))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A2)) ) ) ).

% subset_Diff_insert
tff(fact_3526_subset__Diff__insert,axiom,
    ! [A2: set_nat,B3: set_nat,X3: nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),aa_set_nat_set_nat(insert_nat(X3),C2))))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),C2)))
        & ~ pp(aa_set_nat_bool(member_nat(X3),A2)) ) ) ).

% subset_Diff_insert
tff(fact_3527_subset__Diff__insert,axiom,
    ! [A2: set_int,B3: set_int,X3: int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),aa_set_int_set_int(insert_int(X3),C2))))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),C2)))
        & ~ pp(aa_set_int_bool(member_int(X3),A2)) ) ) ).

% subset_Diff_insert
tff(fact_3528_Collect__conv__if,axiom,
    ! [P: fun_VEBT_VEBT_bool,A: vEBT_VEBT] :
      ( ( pp(aa_VEBT_VEBT_bool(P,A))
       => ( collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_kj(P),A)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ) )
      & ( ~ pp(aa_VEBT_VEBT_bool(P,A))
       => ( collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_kj(P),A)) = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if
tff(fact_3529_Collect__conv__if,axiom,
    ! [P: fun_complex_bool,A: complex] :
      ( ( pp(aa_complex_bool(P,A))
       => ( collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_kk(P),A)) = insert_complex(A,bot_bot_set_complex) ) )
      & ( ~ pp(aa_complex_bool(P,A))
       => ( collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_kk(P),A)) = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
tff(fact_3530_Collect__conv__if,axiom,
    ! [P: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_kl(P),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_kl(P),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
tff(fact_3531_Collect__conv__if,axiom,
    ! [P: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_km(P),A)) = aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_km(P),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_3532_Collect__conv__if,axiom,
    ! [P: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_kn(P),A)) = aa_set_real_set_real(insert_real(A),bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_kn(P),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if
tff(fact_3533_Collect__conv__if,axiom,
    ! [P: fun_bool_bool,A: bool] :
      ( ( pp(aa_bool_bool(P,A))
       => ( collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_ko(P),A)) = aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool) ) )
      & ( ~ pp(aa_bool_bool(P,A))
       => ( collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_ko(P),A)) = bot_bot_set_bool ) ) ) ).

% Collect_conv_if
tff(fact_3534_Collect__conv__if,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_kp(P),A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_kp(P),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
tff(fact_3535_Collect__conv__if,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_kq(P),A)) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_kq(P),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if
tff(fact_3536_Collect__conv__if2,axiom,
    ! [P: fun_VEBT_VEBT_bool,A: vEBT_VEBT] :
      ( ( pp(aa_VEBT_VEBT_bool(P,A))
       => ( collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_kr(P),A)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ) )
      & ( ~ pp(aa_VEBT_VEBT_bool(P,A))
       => ( collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_kr(P),A)) = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if2
tff(fact_3537_Collect__conv__if2,axiom,
    ! [P: fun_complex_bool,A: complex] :
      ( ( pp(aa_complex_bool(P,A))
       => ( collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ks(P),A)) = insert_complex(A,bot_bot_set_complex) ) )
      & ( ~ pp(aa_complex_bool(P,A))
       => ( collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ks(P),A)) = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
tff(fact_3538_Collect__conv__if2,axiom,
    ! [P: fun_list_nat_bool,A: list_nat] :
      ( ( pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_kt(P),A)) = insert_list_nat(A,bot_bot_set_list_nat) ) )
      & ( ~ pp(aa_list_nat_bool(P,A))
       => ( collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_kt(P),A)) = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
tff(fact_3539_Collect__conv__if2,axiom,
    ! [P: fun_set_nat_bool,A: set_nat] :
      ( ( pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_ku(P),A)) = aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat) ) )
      & ( ~ pp(aa_set_nat_bool(P,A))
       => ( collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_ku(P),A)) = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_3540_Collect__conv__if2,axiom,
    ! [P: fun_real_bool,A: real] :
      ( ( pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_kv(P),A)) = aa_set_real_set_real(insert_real(A),bot_bot_set_real) ) )
      & ( ~ pp(aa_real_bool(P,A))
       => ( collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_kv(P),A)) = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
tff(fact_3541_Collect__conv__if2,axiom,
    ! [P: fun_bool_bool,A: bool] :
      ( ( pp(aa_bool_bool(P,A))
       => ( collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_kw(P),A)) = aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool) ) )
      & ( ~ pp(aa_bool_bool(P,A))
       => ( collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_kw(P),A)) = bot_bot_set_bool ) ) ) ).

% Collect_conv_if2
tff(fact_3542_Collect__conv__if2,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ( pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_kx(P),A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) )
      & ( ~ pp(aa_nat_bool(P,A))
       => ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_kx(P),A)) = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
tff(fact_3543_Collect__conv__if2,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ( pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_ky(P),A)) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) )
      & ( ~ pp(aa_int_bool(P,A))
       => ( collect_int(aa_int_fun_int_bool(aTP_Lamm_ky(P),A)) = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
tff(fact_3544_finite_Ocases,axiom,
    ! [A: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
     => ( ( A != bot_bo8194388402131092736T_VEBT )
       => ~ ! [A6: set_VEBT_VEBT] :
              ( ? [A3: vEBT_VEBT] : A = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A3),A6)
             => ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A6)) ) ) ) ).

% finite.cases
tff(fact_3545_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A6: set_complex] :
              ( ? [A3: complex] : A = insert_complex(A3,A6)
             => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A6)) ) ) ) ).

% finite.cases
tff(fact_3546_finite_Ocases,axiom,
    ! [A: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
     => ( ( A != bot_bo3990330152332043303nteger )
       => ~ ! [A6: set_Code_integer] :
              ( ? [A3: code_integer] : A = insert_Code_integer(A3,A6)
             => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A6)) ) ) ) ).

% finite.cases
tff(fact_3547_finite_Ocases,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
     => ( ( A != bot_bot_set_real )
       => ~ ! [A6: set_real] :
              ( ? [A3: real] : A = aa_set_real_set_real(insert_real(A3),A6)
             => ~ pp(aa_set_real_bool(finite_finite_real,A6)) ) ) ) ).

% finite.cases
tff(fact_3548_finite_Ocases,axiom,
    ! [A: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A))
     => ( ( A != bot_bot_set_bool )
       => ~ ! [A6: set_bool] :
              ( ? [A3: bool] : A = aa_set_bool_set_bool(insert_bool(A3),A6)
             => ~ pp(aa_set_bool_bool(finite_finite_bool,A6)) ) ) ) ).

% finite.cases
tff(fact_3549_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A6: set_nat] :
              ( ? [A3: nat] : A = aa_set_nat_set_nat(insert_nat(A3),A6)
             => ~ pp(aa_set_nat_bool(finite_finite_nat,A6)) ) ) ) ).

% finite.cases
tff(fact_3550_finite_Ocases,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
     => ( ( A != bot_bot_set_int )
       => ~ ! [A6: set_int] :
              ( ? [A3: int] : A = aa_set_int_set_int(insert_int(A3),A6)
             => ~ pp(aa_set_int_bool(finite_finite_int,A6)) ) ) ) ).

% finite.cases
tff(fact_3551_finite_Osimps,axiom,
    ! [A: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A))
    <=> ( ( A = bot_bo8194388402131092736T_VEBT )
        | ? [A7: set_VEBT_VEBT,A4: vEBT_VEBT] :
            ( ( A = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A4),A7) )
            & pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A7)) ) ) ) ).

% finite.simps
tff(fact_3552_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A))
    <=> ( ( A = bot_bot_set_complex )
        | ? [A7: set_complex,A4: complex] :
            ( ( A = insert_complex(A4,A7) )
            & pp(aa_set_complex_bool(finite3207457112153483333omplex,A7)) ) ) ) ).

% finite.simps
tff(fact_3553_finite_Osimps,axiom,
    ! [A: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A))
    <=> ( ( A = bot_bo3990330152332043303nteger )
        | ? [A7: set_Code_integer,A4: code_integer] :
            ( ( A = insert_Code_integer(A4,A7) )
            & pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A7)) ) ) ) ).

% finite.simps
tff(fact_3554_finite_Osimps,axiom,
    ! [A: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A))
    <=> ( ( A = bot_bot_set_real )
        | ? [A7: set_real,A4: real] :
            ( ( A = aa_set_real_set_real(insert_real(A4),A7) )
            & pp(aa_set_real_bool(finite_finite_real,A7)) ) ) ) ).

% finite.simps
tff(fact_3555_finite_Osimps,axiom,
    ! [A: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A))
    <=> ( ( A = bot_bot_set_bool )
        | ? [A7: set_bool,A4: bool] :
            ( ( A = aa_set_bool_set_bool(insert_bool(A4),A7) )
            & pp(aa_set_bool_bool(finite_finite_bool,A7)) ) ) ) ).

% finite.simps
tff(fact_3556_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A))
    <=> ( ( A = bot_bot_set_nat )
        | ? [A7: set_nat,A4: nat] :
            ( ( A = aa_set_nat_set_nat(insert_nat(A4),A7) )
            & pp(aa_set_nat_bool(finite_finite_nat,A7)) ) ) ) ).

% finite.simps
tff(fact_3557_finite_Osimps,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A))
    <=> ( ( A = bot_bot_set_int )
        | ? [A7: set_int,A4: int] :
            ( ( A = aa_set_int_set_int(insert_int(A4),A7) )
            & pp(aa_set_int_bool(finite_finite_int,A7)) ) ) ) ).

% finite.simps
tff(fact_3558_finite__induct,axiom,
    ! [F4: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X: vEBT_VEBT,F5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F5))
             => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),F5))
               => ( pp(aa_set8260762808118238015T_bool(P,F5))
                 => pp(aa_set8260762808118238015T_bool(P,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),F5))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,F4)) ) ) ) ).

% finite_induct
tff(fact_3559_finite__induct,axiom,
    ! [F4: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [X: set_nat,F5: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F5))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),F5))
               => ( pp(aa_set_set_nat_bool(P,F5))
                 => pp(aa_set_set_nat_bool(P,aa_set1013257548857649334et_nat(insert_set_nat(X),F5))) ) ) )
         => pp(aa_set_set_nat_bool(P,F4)) ) ) ) ).

% finite_induct
tff(fact_3560_finite__induct,axiom,
    ! [F4: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X: complex,F5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F5))
             => ( ~ pp(aa_set_complex_bool(member_complex(X),F5))
               => ( pp(aa_set_complex_bool(P,F5))
                 => pp(aa_set_complex_bool(P,insert_complex(X,F5))) ) ) )
         => pp(aa_set_complex_bool(P,F4)) ) ) ) ).

% finite_induct
tff(fact_3561_finite__induct,axiom,
    ! [F4: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F4))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [X: code_integer,F5: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F5))
             => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X),F5))
               => ( pp(aa_set1374477347934578310r_bool(P,F5))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X,F5))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,F4)) ) ) ) ).

% finite_induct
tff(fact_3562_finite__induct,axiom,
    ! [F4: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F4))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X: real,F5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F5))
             => ( ~ pp(aa_set_real_bool(member_real(X),F5))
               => ( pp(aa_set_real_bool(P,F5))
                 => pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(X),F5))) ) ) )
         => pp(aa_set_real_bool(P,F4)) ) ) ) ).

% finite_induct
tff(fact_3563_finite__induct,axiom,
    ! [F4: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F4))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [X: bool,F5: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,F5))
             => ( ~ pp(aa_set_bool_bool(member_bool(X),F5))
               => ( pp(aa_set_bool_bool(P,F5))
                 => pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(X),F5))) ) ) )
         => pp(aa_set_bool_bool(P,F4)) ) ) ) ).

% finite_induct
tff(fact_3564_finite__induct,axiom,
    ! [F4: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F4))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X: nat,F5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F5))
             => ( ~ pp(aa_set_nat_bool(member_nat(X),F5))
               => ( pp(aa_set_nat_bool(P,F5))
                 => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(X),F5))) ) ) )
         => pp(aa_set_nat_bool(P,F4)) ) ) ) ).

% finite_induct
tff(fact_3565_finite__induct,axiom,
    ! [F4: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F4))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X: int,F5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F5))
             => ( ~ pp(aa_set_int_bool(member_int(X),F5))
               => ( pp(aa_set_int_bool(P,F5))
                 => pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(X),F5))) ) ) )
         => pp(aa_set_int_bool(P,F4)) ) ) ) ).

% finite_induct
tff(fact_3566_finite__ne__induct,axiom,
    ! [F4: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
     => ( ( F4 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(P,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),bot_bo8194388402131092736T_VEBT)))
         => ( ! [X: vEBT_VEBT,F5: set_VEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F5))
               => ( ( F5 != bot_bo8194388402131092736T_VEBT )
                 => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),F5))
                   => ( pp(aa_set8260762808118238015T_bool(P,F5))
                     => pp(aa_set8260762808118238015T_bool(P,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),F5))) ) ) ) )
           => pp(aa_set8260762808118238015T_bool(P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_3567_finite__ne__induct,axiom,
    ! [F4: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
     => ( ( F4 != bot_bot_set_set_nat )
       => ( ! [X: set_nat] : pp(aa_set_set_nat_bool(P,aa_set1013257548857649334et_nat(insert_set_nat(X),bot_bot_set_set_nat)))
         => ( ! [X: set_nat,F5: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F5))
               => ( ( F5 != bot_bot_set_set_nat )
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),F5))
                   => ( pp(aa_set_set_nat_bool(P,F5))
                     => pp(aa_set_set_nat_bool(P,aa_set1013257548857649334et_nat(insert_set_nat(X),F5))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_3568_finite__ne__induct,axiom,
    ! [F4: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
     => ( ( F4 != bot_bot_set_complex )
       => ( ! [X: complex] : pp(aa_set_complex_bool(P,insert_complex(X,bot_bot_set_complex)))
         => ( ! [X: complex,F5: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F5))
               => ( ( F5 != bot_bot_set_complex )
                 => ( ~ pp(aa_set_complex_bool(member_complex(X),F5))
                   => ( pp(aa_set_complex_bool(P,F5))
                     => pp(aa_set_complex_bool(P,insert_complex(X,F5))) ) ) ) )
           => pp(aa_set_complex_bool(P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_3569_finite__ne__induct,axiom,
    ! [F4: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F4))
     => ( ( F4 != bot_bo3990330152332043303nteger )
       => ( ! [X: code_integer] : pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X,bot_bo3990330152332043303nteger)))
         => ( ! [X: code_integer,F5: set_Code_integer] :
                ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F5))
               => ( ( F5 != bot_bo3990330152332043303nteger )
                 => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X),F5))
                   => ( pp(aa_set1374477347934578310r_bool(P,F5))
                     => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X,F5))) ) ) ) )
           => pp(aa_set1374477347934578310r_bool(P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_3570_finite__ne__induct,axiom,
    ! [F4: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F4))
     => ( ( F4 != bot_bot_set_real )
       => ( ! [X: real] : pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(X),bot_bot_set_real)))
         => ( ! [X: real,F5: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F5))
               => ( ( F5 != bot_bot_set_real )
                 => ( ~ pp(aa_set_real_bool(member_real(X),F5))
                   => ( pp(aa_set_real_bool(P,F5))
                     => pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(X),F5))) ) ) ) )
           => pp(aa_set_real_bool(P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_3571_finite__ne__induct,axiom,
    ! [F4: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F4))
     => ( ( F4 != bot_bot_set_bool )
       => ( ! [X: bool] : pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(X),bot_bot_set_bool)))
         => ( ! [X: bool,F5: set_bool] :
                ( pp(aa_set_bool_bool(finite_finite_bool,F5))
               => ( ( F5 != bot_bot_set_bool )
                 => ( ~ pp(aa_set_bool_bool(member_bool(X),F5))
                   => ( pp(aa_set_bool_bool(P,F5))
                     => pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(X),F5))) ) ) ) )
           => pp(aa_set_bool_bool(P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_3572_finite__ne__induct,axiom,
    ! [F4: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F4))
     => ( ( F4 != bot_bot_set_nat )
       => ( ! [X: nat] : pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)))
         => ( ! [X: nat,F5: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F5))
               => ( ( F5 != bot_bot_set_nat )
                 => ( ~ pp(aa_set_nat_bool(member_nat(X),F5))
                   => ( pp(aa_set_nat_bool(P,F5))
                     => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(X),F5))) ) ) ) )
           => pp(aa_set_nat_bool(P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_3573_finite__ne__induct,axiom,
    ! [F4: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F4))
     => ( ( F4 != bot_bot_set_int )
       => ( ! [X: int] : pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(X),bot_bot_set_int)))
         => ( ! [X: int,F5: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F5))
               => ( ( F5 != bot_bot_set_int )
                 => ( ~ pp(aa_set_int_bool(member_int(X),F5))
                   => ( pp(aa_set_int_bool(P,F5))
                     => pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(X),F5))) ) ) ) )
           => pp(aa_set_int_bool(P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_3574_infinite__finite__induct,axiom,
    ! [P: fun_se448528567104302406T_bool,A2: set_VEBT_VEBT] :
      ( ! [A6: set_VEBT_VEBT] :
          ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A6))
         => pp(aa_set8260762808118238015T_bool(P,A6)) )
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X: vEBT_VEBT,F5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F5))
             => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),F5))
               => ( pp(aa_set8260762808118238015T_bool(P,F5))
                 => pp(aa_set8260762808118238015T_bool(P,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),F5))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_3575_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_bool,A2: set_set_nat] :
      ( ! [A6: set_set_nat] :
          ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A6))
         => pp(aa_set_set_nat_bool(P,A6)) )
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [X: set_nat,F5: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F5))
             => ( ~ pp(aa_set_set_nat_bool(member_set_nat(X),F5))
               => ( pp(aa_set_set_nat_bool(P,F5))
                 => pp(aa_set_set_nat_bool(P,aa_set1013257548857649334et_nat(insert_set_nat(X),F5))) ) ) )
         => pp(aa_set_set_nat_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_3576_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_bool,A2: set_complex] :
      ( ! [A6: set_complex] :
          ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A6))
         => pp(aa_set_complex_bool(P,A6)) )
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X: complex,F5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F5))
             => ( ~ pp(aa_set_complex_bool(member_complex(X),F5))
               => ( pp(aa_set_complex_bool(P,F5))
                 => pp(aa_set_complex_bool(P,insert_complex(X,F5))) ) ) )
         => pp(aa_set_complex_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_3577_infinite__finite__induct,axiom,
    ! [P: fun_se2712834123562403997r_bool,A2: set_Code_integer] :
      ( ! [A6: set_Code_integer] :
          ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A6))
         => pp(aa_set1374477347934578310r_bool(P,A6)) )
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [X: code_integer,F5: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F5))
             => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X),F5))
               => ( pp(aa_set1374477347934578310r_bool(P,F5))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X,F5))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_3578_infinite__finite__induct,axiom,
    ! [P: fun_set_real_bool,A2: set_real] :
      ( ! [A6: set_real] :
          ( ~ pp(aa_set_real_bool(finite_finite_real,A6))
         => pp(aa_set_real_bool(P,A6)) )
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X: real,F5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,F5))
             => ( ~ pp(aa_set_real_bool(member_real(X),F5))
               => ( pp(aa_set_real_bool(P,F5))
                 => pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(X),F5))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_3579_infinite__finite__induct,axiom,
    ! [P: fun_set_bool_bool,A2: set_bool] :
      ( ! [A6: set_bool] :
          ( ~ pp(aa_set_bool_bool(finite_finite_bool,A6))
         => pp(aa_set_bool_bool(P,A6)) )
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [X: bool,F5: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,F5))
             => ( ~ pp(aa_set_bool_bool(member_bool(X),F5))
               => ( pp(aa_set_bool_bool(P,F5))
                 => pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(X),F5))) ) ) )
         => pp(aa_set_bool_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_3580_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_bool,A2: set_nat] :
      ( ! [A6: set_nat] :
          ( ~ pp(aa_set_nat_bool(finite_finite_nat,A6))
         => pp(aa_set_nat_bool(P,A6)) )
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X: nat,F5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,F5))
             => ( ~ pp(aa_set_nat_bool(member_nat(X),F5))
               => ( pp(aa_set_nat_bool(P,F5))
                 => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(X),F5))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_3581_infinite__finite__induct,axiom,
    ! [P: fun_set_int_bool,A2: set_int] :
      ( ! [A6: set_int] :
          ( ~ pp(aa_set_int_bool(finite_finite_int,A6))
         => pp(aa_set_int_bool(P,A6)) )
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X: int,F5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,F5))
             => ( ~ pp(aa_set_int_bool(member_int(X),F5))
               => ( pp(aa_set_int_bool(P,F5))
                 => pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(X),F5))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% infinite_finite_induct
tff(fact_3582_infinite__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(S3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))) ) ).

% infinite_remove
tff(fact_3583_infinite__remove,axiom,
    ! [S3: set_complex,A: complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ).

% infinite_remove
tff(fact_3584_infinite__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,minus_2355218937544613996nteger(S3,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ).

% infinite_remove
tff(fact_3585_infinite__remove,axiom,
    ! [S3: set_real,A: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,S3))
     => ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(S3,aa_set_real_set_real(insert_real(A),bot_bot_set_real)))) ) ).

% infinite_remove
tff(fact_3586_infinite__remove,axiom,
    ! [S3: set_bool,A: bool] :
      ( ~ pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ~ pp(aa_set_bool_bool(finite_finite_bool,minus_minus_set_bool(S3,aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool)))) ) ).

% infinite_remove
tff(fact_3587_infinite__remove,axiom,
    ! [S3: set_int,A: int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
     => ~ pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S3),aa_set_int_set_int(insert_int(A),bot_bot_set_int)))) ) ).

% infinite_remove
tff(fact_3588_infinite__remove,axiom,
    ! [S3: set_nat,A: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ~ pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S3),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))) ) ).

% infinite_remove
tff(fact_3589_infinite__coinduct,axiom,
    ! [X6: fun_se448528567104302406T_bool,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(X6,A2))
     => ( ! [A6: set_VEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(X6,A6))
           => ? [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),A6))
                & ( pp(aa_set8260762808118238015T_bool(X6,minus_5127226145743854075T_VEBT(A6,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X5),bot_bo8194388402131092736T_VEBT))))
                  | ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A6,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X5),bot_bo8194388402131092736T_VEBT)))) ) ) )
       => ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2)) ) ) ).

% infinite_coinduct
tff(fact_3590_infinite__coinduct,axiom,
    ! [X6: fun_set_complex_bool,A2: set_complex] :
      ( pp(aa_set_complex_bool(X6,A2))
     => ( ! [A6: set_complex] :
            ( pp(aa_set_complex_bool(X6,A6))
           => ? [X5: complex] :
                ( pp(aa_set_complex_bool(member_complex(X5),A6))
                & ( pp(aa_set_complex_bool(X6,minus_811609699411566653omplex(A6,insert_complex(X5,bot_bot_set_complex))))
                  | ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A6,insert_complex(X5,bot_bot_set_complex)))) ) ) )
       => ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).

% infinite_coinduct
tff(fact_3591_infinite__coinduct,axiom,
    ! [X6: fun_se2712834123562403997r_bool,A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(X6,A2))
     => ( ! [A6: set_Code_integer] :
            ( pp(aa_set1374477347934578310r_bool(X6,A6))
           => ? [X5: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X5),A6))
                & ( pp(aa_set1374477347934578310r_bool(X6,minus_2355218937544613996nteger(A6,insert_Code_integer(X5,bot_bo3990330152332043303nteger))))
                  | ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,minus_2355218937544613996nteger(A6,insert_Code_integer(X5,bot_bo3990330152332043303nteger)))) ) ) )
       => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2)) ) ) ).

% infinite_coinduct
tff(fact_3592_infinite__coinduct,axiom,
    ! [X6: fun_set_real_bool,A2: set_real] :
      ( pp(aa_set_real_bool(X6,A2))
     => ( ! [A6: set_real] :
            ( pp(aa_set_real_bool(X6,A6))
           => ? [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),A6))
                & ( pp(aa_set_real_bool(X6,minus_minus_set_real(A6,aa_set_real_set_real(insert_real(X5),bot_bot_set_real))))
                  | ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A6,aa_set_real_set_real(insert_real(X5),bot_bot_set_real)))) ) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,A2)) ) ) ).

% infinite_coinduct
tff(fact_3593_infinite__coinduct,axiom,
    ! [X6: fun_set_bool_bool,A2: set_bool] :
      ( pp(aa_set_bool_bool(X6,A2))
     => ( ! [A6: set_bool] :
            ( pp(aa_set_bool_bool(X6,A6))
           => ? [X5: bool] :
                ( pp(aa_set_bool_bool(member_bool(X5),A6))
                & ( pp(aa_set_bool_bool(X6,minus_minus_set_bool(A6,aa_set_bool_set_bool(insert_bool(X5),bot_bot_set_bool))))
                  | ~ pp(aa_set_bool_bool(finite_finite_bool,minus_minus_set_bool(A6,aa_set_bool_set_bool(insert_bool(X5),bot_bot_set_bool)))) ) ) )
       => ~ pp(aa_set_bool_bool(finite_finite_bool,A2)) ) ) ).

% infinite_coinduct
tff(fact_3594_infinite__coinduct,axiom,
    ! [X6: fun_set_int_bool,A2: set_int] :
      ( pp(aa_set_int_bool(X6,A2))
     => ( ! [A6: set_int] :
            ( pp(aa_set_int_bool(X6,A6))
           => ? [X5: int] :
                ( pp(aa_set_int_bool(member_int(X5),A6))
                & ( pp(aa_set_int_bool(X6,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A6),aa_set_int_set_int(insert_int(X5),bot_bot_set_int))))
                  | ~ pp(aa_set_int_bool(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A6),aa_set_int_set_int(insert_int(X5),bot_bot_set_int)))) ) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).

% infinite_coinduct
tff(fact_3595_infinite__coinduct,axiom,
    ! [X6: fun_set_nat_bool,A2: set_nat] :
      ( pp(aa_set_nat_bool(X6,A2))
     => ( ! [A6: set_nat] :
            ( pp(aa_set_nat_bool(X6,A6))
           => ? [X5: nat] :
                ( pp(aa_set_nat_bool(member_nat(X5),A6))
                & ( pp(aa_set_nat_bool(X6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(X5),bot_bot_set_nat))))
                  | ~ pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(X5),bot_bot_set_nat)))) ) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).

% infinite_coinduct
tff(fact_3596_finite__empty__induct,axiom,
    ! [A2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set8260762808118238015T_bool(P,A2))
       => ( ! [A3: vEBT_VEBT,A6: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A6))
             => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A6))
               => ( pp(aa_set8260762808118238015T_bool(P,A6))
                 => pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A6,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A3),bot_bo8194388402131092736T_VEBT)))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT)) ) ) ) ).

% finite_empty_induct
tff(fact_3597_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( pp(aa_set_set_nat_bool(P,A2))
       => ( ! [A3: set_nat,A6: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A6))
             => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A6))
               => ( pp(aa_set_set_nat_bool(P,A6))
                 => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A6,aa_set1013257548857649334et_nat(insert_set_nat(A3),bot_bot_set_set_nat)))) ) ) )
         => pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_3598_finite__empty__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(P,A2))
       => ( ! [A3: complex,A6: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A6))
             => ( pp(aa_set_complex_bool(member_complex(A3),A6))
               => ( pp(aa_set_complex_bool(P,A6))
                 => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A6,insert_complex(A3,bot_bot_set_complex)))) ) ) )
         => pp(aa_set_complex_bool(P,bot_bot_set_complex)) ) ) ) ).

% finite_empty_induct
tff(fact_3599_finite__empty__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(P,A2))
       => ( ! [A3: code_integer,A6: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A6))
             => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A6))
               => ( pp(aa_set1374477347934578310r_bool(P,A6))
                 => pp(aa_set1374477347934578310r_bool(P,minus_2355218937544613996nteger(A6,insert_Code_integer(A3,bot_bo3990330152332043303nteger)))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger)) ) ) ) ).

% finite_empty_induct
tff(fact_3600_finite__empty__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,A2))
       => ( ! [A3: real,A6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A6))
             => ( pp(aa_set_real_bool(member_real(A3),A6))
               => ( pp(aa_set_real_bool(P,A6))
                 => pp(aa_set_real_bool(P,minus_minus_set_real(A6,aa_set_real_set_real(insert_real(A3),bot_bot_set_real)))) ) ) )
         => pp(aa_set_real_bool(P,bot_bot_set_real)) ) ) ) ).

% finite_empty_induct
tff(fact_3601_finite__empty__induct,axiom,
    ! [A2: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( pp(aa_set_bool_bool(P,A2))
       => ( ! [A3: bool,A6: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A6))
             => ( pp(aa_set_bool_bool(member_bool(A3),A6))
               => ( pp(aa_set_bool_bool(P,A6))
                 => pp(aa_set_bool_bool(P,minus_minus_set_bool(A6,aa_set_bool_set_bool(insert_bool(A3),bot_bot_set_bool)))) ) ) )
         => pp(aa_set_bool_bool(P,bot_bot_set_bool)) ) ) ) ).

% finite_empty_induct
tff(fact_3602_finite__empty__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,A2))
       => ( ! [A3: int,A6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A6))
             => ( pp(aa_set_int_bool(member_int(A3),A6))
               => ( pp(aa_set_int_bool(P,A6))
                 => pp(aa_set_int_bool(P,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A6),aa_set_int_set_int(insert_int(A3),bot_bot_set_int)))) ) ) )
         => pp(aa_set_int_bool(P,bot_bot_set_int)) ) ) ) ).

% finite_empty_induct
tff(fact_3603_finite__empty__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,A2))
       => ( ! [A3: nat,A6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A6))
             => ( pp(aa_set_nat_bool(member_nat(A3),A6))
               => ( pp(aa_set_nat_bool(P,A6))
                 => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(A3),bot_bot_set_nat)))) ) ) )
         => pp(aa_set_nat_bool(P,bot_bot_set_nat)) ) ) ) ).

% finite_empty_induct
tff(fact_3604_subset__singleton__iff,axiom,
    ! [X6: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(X6),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))
    <=> ( ( X6 = bot_bo8194388402131092736T_VEBT )
        | ( X6 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singleton_iff
tff(fact_3605_subset__singleton__iff,axiom,
    ! [X6: set_real,A: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),aa_set_real_set_real(insert_real(A),bot_bot_set_real)))
    <=> ( ( X6 = bot_bot_set_real )
        | ( X6 = aa_set_real_set_real(insert_real(A),bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_3606_subset__singleton__iff,axiom,
    ! [X6: set_bool,A: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(X6),aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool)))
    <=> ( ( X6 = bot_bot_set_bool )
        | ( X6 = aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool) ) ) ) ).

% subset_singleton_iff
tff(fact_3607_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))
    <=> ( ( X6 = bot_bot_set_nat )
        | ( X6 = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_3608_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),aa_set_int_set_int(insert_int(A),bot_bot_set_int)))
    <=> ( ( X6 = bot_bot_set_int )
        | ( X6 = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_3609_subset__singletonD,axiom,
    ! [A2: set_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)))
     => ( ( A2 = bot_bo8194388402131092736T_VEBT )
        | ( A2 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singletonD
tff(fact_3610_subset__singletonD,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_3611_subset__singletonD,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)))
     => ( ( A2 = bot_bot_set_bool )
        | ( A2 = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) ) ) ) ).

% subset_singletonD
tff(fact_3612_subset__singletonD,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_3613_subset__singletonD,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_3614_Diff__single__insert,axiom,
    ! [A2: set_VEBT_VEBT,X3: vEBT_VEBT,B3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))),B3))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),B3))) ) ).

% Diff_single_insert
tff(fact_3615_Diff__single__insert,axiom,
    ! [A2: set_real,X3: real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))),B3))
     => pp(aa_set_real_bool(ord_less_eq_set_real(A2),aa_set_real_set_real(insert_real(X3),B3))) ) ).

% Diff_single_insert
tff(fact_3616_Diff__single__insert,axiom,
    ! [A2: set_bool,X3: bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))),B3))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),aa_set_bool_set_bool(insert_bool(X3),B3))) ) ).

% Diff_single_insert
tff(fact_3617_Diff__single__insert,axiom,
    ! [A2: set_nat,X3: nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))),B3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),B3))) ) ).

% Diff_single_insert
tff(fact_3618_Diff__single__insert,axiom,
    ! [A2: set_int,X3: int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int))),B3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(insert_int(X3),B3))) ) ).

% Diff_single_insert
tff(fact_3619_subset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,X3: vEBT_VEBT,B3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),B3)))
    <=> ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
         => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))),B3)) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
         => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3)) ) ) ) ).

% subset_insert_iff
tff(fact_3620_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X3: set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),aa_set1013257548857649334et_nat(insert_set_nat(X3),B3)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat))),B3)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
         => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3)) ) ) ) ).

% subset_insert_iff
tff(fact_3621_subset__insert__iff,axiom,
    ! [A2: set_real,X3: real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),aa_set_real_set_real(insert_real(X3),B3)))
    <=> ( ( pp(aa_set_real_bool(member_real(X3),A2))
         => pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))),B3)) )
        & ( ~ pp(aa_set_real_bool(member_real(X3),A2))
         => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3)) ) ) ) ).

% subset_insert_iff
tff(fact_3622_subset__insert__iff,axiom,
    ! [A2: set_bool,X3: bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),aa_set_bool_set_bool(insert_bool(X3),B3)))
    <=> ( ( pp(aa_set_bool_bool(member_bool(X3),A2))
         => pp(aa_set_bool_bool(ord_less_eq_set_bool(minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))),B3)) )
        & ( ~ pp(aa_set_bool_bool(member_bool(X3),A2))
         => pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),B3)) ) ) ) ).

% subset_insert_iff
tff(fact_3623_subset__insert__iff,axiom,
    ! [A2: set_nat,X3: nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),B3)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))),B3)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X3),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3)) ) ) ) ).

% subset_insert_iff
tff(fact_3624_subset__insert__iff,axiom,
    ! [A2: set_int,X3: int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(insert_int(X3),B3)))
    <=> ( ( pp(aa_set_int_bool(member_int(X3),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int))),B3)) )
        & ( ~ pp(aa_set_int_bool(member_int(X3),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ) ) ).

% subset_insert_iff
tff(fact_3625_remove__subset,axiom,
    ! [X3: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),S3))
     => pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(S3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))),S3)) ) ).

% remove_subset
tff(fact_3626_remove__subset,axiom,
    ! [X3: set_nat,S3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),S3))
     => pp(aa_set_set_nat_bool(ord_less_set_set_nat(minus_2163939370556025621et_nat(S3,aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat))),S3)) ) ).

% remove_subset
tff(fact_3627_remove__subset,axiom,
    ! [X3: real,S3: set_real] :
      ( pp(aa_set_real_bool(member_real(X3),S3))
     => pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(S3,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))),S3)) ) ).

% remove_subset
tff(fact_3628_remove__subset,axiom,
    ! [X3: bool,S3: set_bool] :
      ( pp(aa_set_bool_bool(member_bool(X3),S3))
     => pp(aa_set_bool_bool(ord_less_set_bool(minus_minus_set_bool(S3,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))),S3)) ) ).

% remove_subset
tff(fact_3629_remove__subset,axiom,
    ! [X3: int,S3: set_int] :
      ( pp(aa_set_int_bool(member_int(X3),S3))
     => pp(aa_set_int_bool(ord_less_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S3),aa_set_int_set_int(insert_int(X3),bot_bot_set_int))),S3)) ) ).

% remove_subset
tff(fact_3630_remove__subset,axiom,
    ! [X3: nat,S3: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),S3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S3),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))),S3)) ) ).

% remove_subset
tff(fact_3631_product__concat__map,axiom,
    ! [Xs: list_nat,Ys: list_nat] : product_nat_nat(Xs,Ys) = concat7691415812945658306at_nat(map_na4561905831291441265at_nat(aTP_Lamm_kz(Ys),Xs)) ).

% product_concat_map
tff(fact_3632_product__concat__map,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : produc7295137177222721919BT_nat(Xs,Ys) = concat4407583305730369330BT_nat(map_VE4456067000678548787BT_nat(aTP_Lamm_la(Ys),Xs)) ).

% product_concat_map
tff(fact_3633_product__concat__map,axiom,
    ! [Xs: list_int,Ys: list_int] : product_int_int(Xs,Ys) = concat4512918505337516154nt_int(map_in7266296235447420877nt_int(aTP_Lamm_lb(Ys),Xs)) ).

% product_concat_map
tff(fact_3634_product__concat__map,axiom,
    ! [Xs: list_Code_integer,Ys: list_Code_integer] : produc8792966785426426881nteger(Xs,Ys) = concat6978052072357435484nteger(map_Co2570546026900698046nteger(aTP_Lamm_lc(Ys),Xs)) ).

% product_concat_map
tff(fact_3635_product__concat__map,axiom,
    ! [Xs: list_nat,Ys: list_num] : product_nat_num(Xs,Ys) = concat8399659749894496012at_num(map_na277125420730788027at_num(aTP_Lamm_ld(Ys),Xs)) ).

% product_concat_map
tff(fact_3636_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: fun_se448528567104302406T_bool,F: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S5))
             => ( ! [Y4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,Y4)),aa_VEBT_VEBT_rat(F,X))) )
               => ( pp(aa_set8260762808118238015T_bool(P,S5))
                 => pp(aa_set8260762808118238015T_bool(P,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),S5))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3637_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_bool,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X: complex,S5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,Y4)),aa_complex_rat(F,X))) )
               => ( pp(aa_set_complex_bool(P,S5))
                 => pp(aa_set_complex_bool(P,insert_complex(X,S5))) ) ) )
         => pp(aa_set_complex_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3638_finite__ranking__induct,axiom,
    ! [S3: set_Code_integer,P: fun_se2712834123562403997r_bool,F: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [X: code_integer,S5: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S5))
             => ( ! [Y4: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(F,Y4)),aa_Code_integer_rat(F,X))) )
               => ( pp(aa_set1374477347934578310r_bool(P,S5))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X,S5))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3639_finite__ranking__induct,axiom,
    ! [S3: set_real,P: fun_set_real_bool,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [X: real,S5: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,S5))
             => ( ! [Y4: real] :
                    ( pp(aa_set_real_bool(member_real(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,Y4)),aa_real_rat(F,X))) )
               => ( pp(aa_set_real_bool(P,S5))
                 => pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(X),S5))) ) ) )
         => pp(aa_set_real_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3640_finite__ranking__induct,axiom,
    ! [S3: set_bool,P: fun_set_bool_bool,F: fun_bool_rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [X: bool,S5: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,S5))
             => ( ! [Y4: bool] :
                    ( pp(aa_set_bool_bool(member_bool(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,Y4)),aa_bool_rat(F,X))) )
               => ( pp(aa_set_bool_bool(P,S5))
                 => pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(X),S5))) ) ) )
         => pp(aa_set_bool_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3641_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: fun_set_nat_bool,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [X: nat,S5: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,S5))
             => ( ! [Y4: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X))) )
               => ( pp(aa_set_nat_bool(P,S5))
                 => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(X),S5))) ) ) )
         => pp(aa_set_nat_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3642_finite__ranking__induct,axiom,
    ! [S3: set_int,P: fun_set_int_bool,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [X: int,S5: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,S5))
             => ( ! [Y4: int] :
                    ( pp(aa_set_int_bool(member_int(Y4),S5))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X))) )
               => ( pp(aa_set_int_bool(P,S5))
                 => pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(X),S5))) ) ) )
         => pp(aa_set_int_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3643_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: fun_se448528567104302406T_bool,F: fun_VEBT_VEBT_num] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [X: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S5))
             => ( ! [Y4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y4),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_VEBT_VEBT_num(F,Y4)),aa_VEBT_VEBT_num(F,X))) )
               => ( pp(aa_set8260762808118238015T_bool(P,S5))
                 => pp(aa_set8260762808118238015T_bool(P,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),S5))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3644_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_bool,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [X: complex,S5: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
             => ( ! [Y4: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Y4),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,Y4)),aa_complex_num(F,X))) )
               => ( pp(aa_set_complex_bool(P,S5))
                 => pp(aa_set_complex_bool(P,insert_complex(X,S5))) ) ) )
         => pp(aa_set_complex_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3645_finite__ranking__induct,axiom,
    ! [S3: set_Code_integer,P: fun_se2712834123562403997r_bool,F: fun_Code_integer_num] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [X: code_integer,S5: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S5))
             => ( ! [Y4: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Y4),S5))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_Code_integer_num(F,Y4)),aa_Code_integer_num(F,X))) )
               => ( pp(aa_set1374477347934578310r_bool(P,S5))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X,S5))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,S3)) ) ) ) ).

% finite_ranking_induct
tff(fact_3646_finite__linorder__max__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [B2: code_integer,A6: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A6))
             => ( ! [X5: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X5),A6))
                   => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X5),B2)) )
               => ( pp(aa_set1374477347934578310r_bool(P,A6))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(B2,A6))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3647_finite__linorder__max__induct,axiom,
    ! [A2: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [B2: bool,A6: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A6))
             => ( ! [X5: bool] :
                    ( pp(aa_set_bool_bool(member_bool(X5),A6))
                   => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X5),B2)) )
               => ( pp(aa_set_bool_bool(P,A6))
                 => pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(B2),A6))) ) ) )
         => pp(aa_set_bool_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3648_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B2: real,A6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A6))
             => ( ! [X5: real] :
                    ( pp(aa_set_real_bool(member_real(X5),A6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),B2)) )
               => ( pp(aa_set_real_bool(P,A6))
                 => pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(B2),A6))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3649_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B2: rat,A6: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A6))
             => ( ! [X5: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X5),A6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),B2)) )
               => ( pp(aa_set_rat_bool(P,A6))
                 => pp(aa_set_rat_bool(P,insert_rat(B2,A6))) ) ) )
         => pp(aa_set_rat_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3650_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B2: num,A6: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A6))
             => ( ! [X5: num] :
                    ( pp(aa_set_num_bool(member_num(X5),A6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X5),B2)) )
               => ( pp(aa_set_num_bool(P,A6))
                 => pp(aa_set_num_bool(P,insert_num(B2,A6))) ) ) )
         => pp(aa_set_num_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3651_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B2: nat,A6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A6))
             => ( ! [X5: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X5),A6))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),B2)) )
               => ( pp(aa_set_nat_bool(P,A6))
                 => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(B2),A6))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3652_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B2: int,A6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A6))
             => ( ! [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),A6))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),B2)) )
               => ( pp(aa_set_int_bool(P,A6))
                 => pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(B2),A6))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3653_finite__linorder__min__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [B2: code_integer,A6: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A6))
             => ( ! [X5: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X5),A6))
                   => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),X5)) )
               => ( pp(aa_set1374477347934578310r_bool(P,A6))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(B2,A6))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3654_finite__linorder__min__induct,axiom,
    ! [A2: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [B2: bool,A6: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A6))
             => ( ! [X5: bool] :
                    ( pp(aa_set_bool_bool(member_bool(X5),A6))
                   => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,B2),X5)) )
               => ( pp(aa_set_bool_bool(P,A6))
                 => pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(B2),A6))) ) ) )
         => pp(aa_set_bool_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3655_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B2: real,A6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A6))
             => ( ! [X5: real] :
                    ( pp(aa_set_real_bool(member_real(X5),A6))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B2),X5)) )
               => ( pp(aa_set_real_bool(P,A6))
                 => pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(B2),A6))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3656_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B2: rat,A6: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A6))
             => ( ! [X5: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X5),A6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B2),X5)) )
               => ( pp(aa_set_rat_bool(P,A6))
                 => pp(aa_set_rat_bool(P,insert_rat(B2,A6))) ) ) )
         => pp(aa_set_rat_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3657_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B2: num,A6: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A6))
             => ( ! [X5: num] :
                    ( pp(aa_set_num_bool(member_num(X5),A6))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),X5)) )
               => ( pp(aa_set_num_bool(P,A6))
                 => pp(aa_set_num_bool(P,insert_num(B2,A6))) ) ) )
         => pp(aa_set_num_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3658_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B2: nat,A6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A6))
             => ( ! [X5: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X5),A6))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),X5)) )
               => ( pp(aa_set_nat_bool(P,A6))
                 => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(B2),A6))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3659_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B2: int,A6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A6))
             => ( ! [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),A6))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),X5)) )
               => ( pp(aa_set_int_bool(P,A6))
                 => pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(B2),A6))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3660_finite__subset__induct,axiom,
    ! [F4: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(F4),A2))
       => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
         => ( ! [A3: vEBT_VEBT,F5: set_VEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F5))
               => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A2))
                 => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),F5))
                   => ( pp(aa_set8260762808118238015T_bool(P,F5))
                     => pp(aa_set8260762808118238015T_bool(P,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A3),F5))) ) ) ) )
           => pp(aa_set8260762808118238015T_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3661_finite__subset__induct,axiom,
    ! [F4: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F4),A2))
       => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
         => ( ! [A3: set_nat,F5: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F5))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A2))
                 => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F5))
                   => ( pp(aa_set_set_nat_bool(P,F5))
                     => pp(aa_set_set_nat_bool(P,aa_set1013257548857649334et_nat(insert_set_nat(A3),F5))) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3662_finite__subset__induct,axiom,
    ! [F4: set_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F4),A2))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F5: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F5))
               => ( pp(aa_set_complex_bool(member_complex(A3),A2))
                 => ( ~ pp(aa_set_complex_bool(member_complex(A3),F5))
                   => ( pp(aa_set_complex_bool(P,F5))
                     => pp(aa_set_complex_bool(P,insert_complex(A3,F5))) ) ) ) )
           => pp(aa_set_complex_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3663_finite__subset__induct,axiom,
    ! [F4: set_Code_integer,A2: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F4))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(F4),A2))
       => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
         => ( ! [A3: code_integer,F5: set_Code_integer] :
                ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F5))
               => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A2))
                 => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),F5))
                   => ( pp(aa_set1374477347934578310r_bool(P,F5))
                     => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(A3,F5))) ) ) ) )
           => pp(aa_set1374477347934578310r_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3664_finite__subset__induct,axiom,
    ! [F4: set_real,A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F4),A2))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A3: real,F5: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F5))
               => ( pp(aa_set_real_bool(member_real(A3),A2))
                 => ( ~ pp(aa_set_real_bool(member_real(A3),F5))
                   => ( pp(aa_set_real_bool(P,F5))
                     => pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(A3),F5))) ) ) ) )
           => pp(aa_set_real_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3665_finite__subset__induct,axiom,
    ! [F4: set_bool,A2: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F4))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(F4),A2))
       => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
         => ( ! [A3: bool,F5: set_bool] :
                ( pp(aa_set_bool_bool(finite_finite_bool,F5))
               => ( pp(aa_set_bool_bool(member_bool(A3),A2))
                 => ( ~ pp(aa_set_bool_bool(member_bool(A3),F5))
                   => ( pp(aa_set_bool_bool(P,F5))
                     => pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(A3),F5))) ) ) ) )
           => pp(aa_set_bool_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3666_finite__subset__induct,axiom,
    ! [F4: set_nat,A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F4),A2))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A3: nat,F5: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F5))
               => ( pp(aa_set_nat_bool(member_nat(A3),A2))
                 => ( ~ pp(aa_set_nat_bool(member_nat(A3),F5))
                   => ( pp(aa_set_nat_bool(P,F5))
                     => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(A3),F5))) ) ) ) )
           => pp(aa_set_nat_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3667_finite__subset__induct,axiom,
    ! [F4: set_int,A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F4),A2))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A3: int,F5: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F5))
               => ( pp(aa_set_int_bool(member_int(A3),A2))
                 => ( ~ pp(aa_set_int_bool(member_int(A3),F5))
                   => ( pp(aa_set_int_bool(P,F5))
                     => pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(A3),F5))) ) ) ) )
           => pp(aa_set_int_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3668_finite__subset__induct_H,axiom,
    ! [F4: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(F4),A2))
       => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
         => ( ! [A3: vEBT_VEBT,F5: set_VEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F5))
               => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A2))
                 => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(F5),A2))
                   => ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),F5))
                     => ( pp(aa_set8260762808118238015T_bool(P,F5))
                       => pp(aa_set8260762808118238015T_bool(P,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A3),F5))) ) ) ) ) )
           => pp(aa_set8260762808118238015T_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3669_finite__subset__induct_H,axiom,
    ! [F4: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F4))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F4),A2))
       => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
         => ( ! [A3: set_nat,F5: set_set_nat] :
                ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F5))
               => ( pp(aa_set_set_nat_bool(member_set_nat(A3),A2))
                 => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F5),A2))
                   => ( ~ pp(aa_set_set_nat_bool(member_set_nat(A3),F5))
                     => ( pp(aa_set_set_nat_bool(P,F5))
                       => pp(aa_set_set_nat_bool(P,aa_set1013257548857649334et_nat(insert_set_nat(A3),F5))) ) ) ) ) )
           => pp(aa_set_set_nat_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3670_finite__subset__induct_H,axiom,
    ! [F4: set_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F4),A2))
       => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
         => ( ! [A3: complex,F5: set_complex] :
                ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F5))
               => ( pp(aa_set_complex_bool(member_complex(A3),A2))
                 => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(F5),A2))
                   => ( ~ pp(aa_set_complex_bool(member_complex(A3),F5))
                     => ( pp(aa_set_complex_bool(P,F5))
                       => pp(aa_set_complex_bool(P,insert_complex(A3,F5))) ) ) ) ) )
           => pp(aa_set_complex_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3671_finite__subset__induct_H,axiom,
    ! [F4: set_Code_integer,A2: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F4))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(F4),A2))
       => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
         => ( ! [A3: code_integer,F5: set_Code_integer] :
                ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F5))
               => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A2))
                 => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(F5),A2))
                   => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),F5))
                     => ( pp(aa_set1374477347934578310r_bool(P,F5))
                       => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(A3,F5))) ) ) ) ) )
           => pp(aa_set1374477347934578310r_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3672_finite__subset__induct_H,axiom,
    ! [F4: set_real,A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,F4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(F4),A2))
       => ( pp(aa_set_real_bool(P,bot_bot_set_real))
         => ( ! [A3: real,F5: set_real] :
                ( pp(aa_set_real_bool(finite_finite_real,F5))
               => ( pp(aa_set_real_bool(member_real(A3),A2))
                 => ( pp(aa_set_real_bool(ord_less_eq_set_real(F5),A2))
                   => ( ~ pp(aa_set_real_bool(member_real(A3),F5))
                     => ( pp(aa_set_real_bool(P,F5))
                       => pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(A3),F5))) ) ) ) ) )
           => pp(aa_set_real_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3673_finite__subset__induct_H,axiom,
    ! [F4: set_bool,A2: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F4))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(F4),A2))
       => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
         => ( ! [A3: bool,F5: set_bool] :
                ( pp(aa_set_bool_bool(finite_finite_bool,F5))
               => ( pp(aa_set_bool_bool(member_bool(A3),A2))
                 => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(F5),A2))
                   => ( ~ pp(aa_set_bool_bool(member_bool(A3),F5))
                     => ( pp(aa_set_bool_bool(P,F5))
                       => pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(A3),F5))) ) ) ) ) )
           => pp(aa_set_bool_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3674_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F4))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F4),A2))
       => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
         => ( ! [A3: nat,F5: set_nat] :
                ( pp(aa_set_nat_bool(finite_finite_nat,F5))
               => ( pp(aa_set_nat_bool(member_nat(A3),A2))
                 => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,F5),A2))
                   => ( ~ pp(aa_set_nat_bool(member_nat(A3),F5))
                     => ( pp(aa_set_nat_bool(P,F5))
                       => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(A3),F5))) ) ) ) ) )
           => pp(aa_set_nat_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3675_finite__subset__induct_H,axiom,
    ! [F4: set_int,A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,F4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(F4),A2))
       => ( pp(aa_set_int_bool(P,bot_bot_set_int))
         => ( ! [A3: int,F5: set_int] :
                ( pp(aa_set_int_bool(finite_finite_int,F5))
               => ( pp(aa_set_int_bool(member_int(A3),A2))
                 => ( pp(aa_set_int_bool(ord_less_eq_set_int(F5),A2))
                   => ( ~ pp(aa_set_int_bool(member_int(A3),F5))
                     => ( pp(aa_set_int_bool(P,F5))
                       => pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(A3),F5))) ) ) ) ) )
           => pp(aa_set_int_bool(P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3676_remove__induct,axiom,
    ! [P: fun_se448528567104302406T_bool,B3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
     => ( ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B3))
         => pp(aa_set8260762808118238015T_bool(P,B3)) )
       => ( ! [A6: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A6))
             => ( ( A6 != bot_bo8194388402131092736T_VEBT )
               => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A6),B3))
                 => ( ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),A6))
                       => pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A6,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X5),bot_bo8194388402131092736T_VEBT)))) )
                   => pp(aa_set8260762808118238015T_bool(P,A6)) ) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,B3)) ) ) ) ).

% remove_induct
tff(fact_3677_remove__induct,axiom,
    ! [P: fun_set_set_nat_bool,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
     => ( ( ~ pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B3))
         => pp(aa_set_set_nat_bool(P,B3)) )
       => ( ! [A6: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A6))
             => ( ( A6 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A6),B3))
                 => ( ! [X5: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X5),A6))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A6,aa_set1013257548857649334et_nat(insert_set_nat(X5),bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A6)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B3)) ) ) ) ).

% remove_induct
tff(fact_3678_remove__induct,axiom,
    ! [P: fun_set_complex_bool,B3: set_complex] :
      ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
     => ( ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
         => pp(aa_set_complex_bool(P,B3)) )
       => ( ! [A6: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A6))
             => ( ( A6 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A6),B3))
                 => ( ! [X5: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X5),A6))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A6,insert_complex(X5,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A6)) ) ) ) )
         => pp(aa_set_complex_bool(P,B3)) ) ) ) ).

% remove_induct
tff(fact_3679_remove__induct,axiom,
    ! [P: fun_se2712834123562403997r_bool,B3: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
     => ( ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
         => pp(aa_set1374477347934578310r_bool(P,B3)) )
       => ( ! [A6: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A6))
             => ( ( A6 != bot_bo3990330152332043303nteger )
               => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A6),B3))
                 => ( ! [X5: code_integer] :
                        ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X5),A6))
                       => pp(aa_set1374477347934578310r_bool(P,minus_2355218937544613996nteger(A6,insert_Code_integer(X5,bot_bo3990330152332043303nteger)))) )
                   => pp(aa_set1374477347934578310r_bool(P,A6)) ) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,B3)) ) ) ) ).

% remove_induct
tff(fact_3680_remove__induct,axiom,
    ! [P: fun_set_real_bool,B3: set_real] :
      ( pp(aa_set_real_bool(P,bot_bot_set_real))
     => ( ( ~ pp(aa_set_real_bool(finite_finite_real,B3))
         => pp(aa_set_real_bool(P,B3)) )
       => ( ! [A6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A6))
             => ( ( A6 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A6),B3))
                 => ( ! [X5: real] :
                        ( pp(aa_set_real_bool(member_real(X5),A6))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A6,aa_set_real_set_real(insert_real(X5),bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A6)) ) ) ) )
         => pp(aa_set_real_bool(P,B3)) ) ) ) ).

% remove_induct
tff(fact_3681_remove__induct,axiom,
    ! [P: fun_set_bool_bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
     => ( ( ~ pp(aa_set_bool_bool(finite_finite_bool,B3))
         => pp(aa_set_bool_bool(P,B3)) )
       => ( ! [A6: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A6))
             => ( ( A6 != bot_bot_set_bool )
               => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A6),B3))
                 => ( ! [X5: bool] :
                        ( pp(aa_set_bool_bool(member_bool(X5),A6))
                       => pp(aa_set_bool_bool(P,minus_minus_set_bool(A6,aa_set_bool_set_bool(insert_bool(X5),bot_bot_set_bool)))) )
                   => pp(aa_set_bool_bool(P,A6)) ) ) ) )
         => pp(aa_set_bool_bool(P,B3)) ) ) ) ).

% remove_induct
tff(fact_3682_remove__induct,axiom,
    ! [P: fun_set_nat_bool,B3: set_nat] :
      ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
     => ( ( ~ pp(aa_set_nat_bool(finite_finite_nat,B3))
         => pp(aa_set_nat_bool(P,B3)) )
       => ( ! [A6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A6))
             => ( ( A6 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A6),B3))
                 => ( ! [X5: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X5),A6))
                       => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(X5),bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A6)) ) ) ) )
         => pp(aa_set_nat_bool(P,B3)) ) ) ) ).

% remove_induct
tff(fact_3683_remove__induct,axiom,
    ! [P: fun_set_int_bool,B3: set_int] :
      ( pp(aa_set_int_bool(P,bot_bot_set_int))
     => ( ( ~ pp(aa_set_int_bool(finite_finite_int,B3))
         => pp(aa_set_int_bool(P,B3)) )
       => ( ! [A6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A6))
             => ( ( A6 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A6),B3))
                 => ( ! [X5: int] :
                        ( pp(aa_set_int_bool(member_int(X5),A6))
                       => pp(aa_set_int_bool(P,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A6),aa_set_int_set_int(insert_int(X5),bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A6)) ) ) ) )
         => pp(aa_set_int_bool(P,B3)) ) ) ) ).

% remove_induct
tff(fact_3684_finite__remove__induct,axiom,
    ! [B3: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B3))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [A6: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A6))
             => ( ( A6 != bot_bo8194388402131092736T_VEBT )
               => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A6),B3))
                 => ( ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),A6))
                       => pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A6,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X5),bot_bo8194388402131092736T_VEBT)))) )
                   => pp(aa_set8260762808118238015T_bool(P,A6)) ) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_3685_finite__remove__induct,axiom,
    ! [B3: set_set_nat,P: fun_set_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,B3))
     => ( pp(aa_set_set_nat_bool(P,bot_bot_set_set_nat))
       => ( ! [A6: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A6))
             => ( ( A6 != bot_bot_set_set_nat )
               => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A6),B3))
                 => ( ! [X5: set_nat] :
                        ( pp(aa_set_set_nat_bool(member_set_nat(X5),A6))
                       => pp(aa_set_set_nat_bool(P,minus_2163939370556025621et_nat(A6,aa_set1013257548857649334et_nat(insert_set_nat(X5),bot_bot_set_set_nat)))) )
                   => pp(aa_set_set_nat_bool(P,A6)) ) ) ) )
         => pp(aa_set_set_nat_bool(P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_3686_finite__remove__induct,axiom,
    ! [B3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [A6: set_complex] :
              ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A6))
             => ( ( A6 != bot_bot_set_complex )
               => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A6),B3))
                 => ( ! [X5: complex] :
                        ( pp(aa_set_complex_bool(member_complex(X5),A6))
                       => pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A6,insert_complex(X5,bot_bot_set_complex)))) )
                   => pp(aa_set_complex_bool(P,A6)) ) ) ) )
         => pp(aa_set_complex_bool(P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_3687_finite__remove__induct,axiom,
    ! [B3: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [A6: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A6))
             => ( ( A6 != bot_bo3990330152332043303nteger )
               => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A6),B3))
                 => ( ! [X5: code_integer] :
                        ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X5),A6))
                       => pp(aa_set1374477347934578310r_bool(P,minus_2355218937544613996nteger(A6,insert_Code_integer(X5,bot_bo3990330152332043303nteger)))) )
                   => pp(aa_set1374477347934578310r_bool(P,A6)) ) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_3688_finite__remove__induct,axiom,
    ! [B3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,B3))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [A6: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A6))
             => ( ( A6 != bot_bot_set_real )
               => ( pp(aa_set_real_bool(ord_less_eq_set_real(A6),B3))
                 => ( ! [X5: real] :
                        ( pp(aa_set_real_bool(member_real(X5),A6))
                       => pp(aa_set_real_bool(P,minus_minus_set_real(A6,aa_set_real_set_real(insert_real(X5),bot_bot_set_real)))) )
                   => pp(aa_set_real_bool(P,A6)) ) ) ) )
         => pp(aa_set_real_bool(P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_3689_finite__remove__induct,axiom,
    ! [B3: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,B3))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [A6: set_bool] :
              ( pp(aa_set_bool_bool(finite_finite_bool,A6))
             => ( ( A6 != bot_bot_set_bool )
               => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A6),B3))
                 => ( ! [X5: bool] :
                        ( pp(aa_set_bool_bool(member_bool(X5),A6))
                       => pp(aa_set_bool_bool(P,minus_minus_set_bool(A6,aa_set_bool_set_bool(insert_bool(X5),bot_bot_set_bool)))) )
                   => pp(aa_set_bool_bool(P,A6)) ) ) ) )
         => pp(aa_set_bool_bool(P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_3690_finite__remove__induct,axiom,
    ! [B3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B3))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [A6: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A6))
             => ( ( A6 != bot_bot_set_nat )
               => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A6),B3))
                 => ( ! [X5: nat] :
                        ( pp(aa_set_nat_bool(member_nat(X5),A6))
                       => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A6),aa_set_nat_set_nat(insert_nat(X5),bot_bot_set_nat)))) )
                   => pp(aa_set_nat_bool(P,A6)) ) ) ) )
         => pp(aa_set_nat_bool(P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_3691_finite__remove__induct,axiom,
    ! [B3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,B3))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [A6: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A6))
             => ( ( A6 != bot_bot_set_int )
               => ( pp(aa_set_int_bool(ord_less_eq_set_int(A6),B3))
                 => ( ! [X5: int] :
                        ( pp(aa_set_int_bool(member_int(X5),A6))
                       => pp(aa_set_int_bool(P,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A6),aa_set_int_set_int(insert_int(X5),bot_bot_set_int)))) )
                   => pp(aa_set_int_bool(P,A6)) ) ) ) )
         => pp(aa_set_int_bool(P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_3692_finite__induct__select,axiom,
    ! [S3: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
       => ( ! [T5: set_VEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(T5),S3))
             => ( pp(aa_set8260762808118238015T_bool(P,T5))
               => ? [X5: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),minus_5127226145743854075T_VEBT(S3,T5)))
                    & pp(aa_set8260762808118238015T_bool(P,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X5),T5))) ) ) )
         => pp(aa_set8260762808118238015T_bool(P,S3)) ) ) ) ).

% finite_induct_select
tff(fact_3693_finite__induct__select,axiom,
    ! [S3: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
       => ( ! [T5: set_complex] :
              ( pp(aa_set_complex_bool(ord_less_set_complex(T5),S3))
             => ( pp(aa_set_complex_bool(P,T5))
               => ? [X5: complex] :
                    ( pp(aa_set_complex_bool(member_complex(X5),minus_811609699411566653omplex(S3,T5)))
                    & pp(aa_set_complex_bool(P,insert_complex(X5,T5))) ) ) )
         => pp(aa_set_complex_bool(P,S3)) ) ) ) ).

% finite_induct_select
tff(fact_3694_finite__induct__select,axiom,
    ! [S3: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [T5: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(ord_le1307284697595431911nteger(T5),S3))
             => ( pp(aa_set1374477347934578310r_bool(P,T5))
               => ? [X5: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X5),minus_2355218937544613996nteger(S3,T5)))
                    & pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(X5,T5))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,S3)) ) ) ) ).

% finite_induct_select
tff(fact_3695_finite__induct__select,axiom,
    ! [S3: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [T5: set_real] :
              ( pp(aa_set_real_bool(ord_less_set_real(T5),S3))
             => ( pp(aa_set_real_bool(P,T5))
               => ? [X5: real] :
                    ( pp(aa_set_real_bool(member_real(X5),minus_minus_set_real(S3,T5)))
                    & pp(aa_set_real_bool(P,aa_set_real_set_real(insert_real(X5),T5))) ) ) )
         => pp(aa_set_real_bool(P,S3)) ) ) ) ).

% finite_induct_select
tff(fact_3696_finite__induct__select,axiom,
    ! [S3: set_bool,P: fun_set_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ( pp(aa_set_bool_bool(P,bot_bot_set_bool))
       => ( ! [T5: set_bool] :
              ( pp(aa_set_bool_bool(ord_less_set_bool(T5),S3))
             => ( pp(aa_set_bool_bool(P,T5))
               => ? [X5: bool] :
                    ( pp(aa_set_bool_bool(member_bool(X5),minus_minus_set_bool(S3,T5)))
                    & pp(aa_set_bool_bool(P,aa_set_bool_set_bool(insert_bool(X5),T5))) ) ) )
         => pp(aa_set_bool_bool(P,S3)) ) ) ) ).

% finite_induct_select
tff(fact_3697_finite__induct__select,axiom,
    ! [S3: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [T5: set_int] :
              ( pp(aa_set_int_bool(ord_less_set_int(T5),S3))
             => ( pp(aa_set_int_bool(P,T5))
               => ? [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S3),T5)))
                    & pp(aa_set_int_bool(P,aa_set_int_set_int(insert_int(X5),T5))) ) ) )
         => pp(aa_set_int_bool(P,S3)) ) ) ) ).

% finite_induct_select
tff(fact_3698_finite__induct__select,axiom,
    ! [S3: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [T5: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,T5),S3))
             => ( pp(aa_set_nat_bool(P,T5))
               => ? [X5: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S3),T5)))
                    & pp(aa_set_nat_bool(P,aa_set_nat_set_nat(insert_nat(X5),T5))) ) ) )
         => pp(aa_set_nat_bool(P,S3)) ) ) ) ).

% finite_induct_select
tff(fact_3699_psubset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,X3: vEBT_VEBT,B3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),B3)))
    <=> ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),B3))
         => pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A2),B3)) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),B3))
         => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
             => pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))),B3)) )
            & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
             => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_3700_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X3: set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),aa_set1013257548857649334et_nat(insert_set_nat(X3),B3)))
    <=> ( ( pp(aa_set_set_nat_bool(member_set_nat(X3),B3))
         => pp(aa_set_set_nat_bool(ord_less_set_set_nat(A2),B3)) )
        & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),B3))
         => ( ( pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
             => pp(aa_set_set_nat_bool(ord_less_set_set_nat(minus_2163939370556025621et_nat(A2,aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat))),B3)) )
            & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
             => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_3701_psubset__insert__iff,axiom,
    ! [A2: set_real,X3: real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_set_real(A2),aa_set_real_set_real(insert_real(X3),B3)))
    <=> ( ( pp(aa_set_real_bool(member_real(X3),B3))
         => pp(aa_set_real_bool(ord_less_set_real(A2),B3)) )
        & ( ~ pp(aa_set_real_bool(member_real(X3),B3))
         => ( ( pp(aa_set_real_bool(member_real(X3),A2))
             => pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))),B3)) )
            & ( ~ pp(aa_set_real_bool(member_real(X3),A2))
             => pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_3702_psubset__insert__iff,axiom,
    ! [A2: set_bool,X3: bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_set_bool(A2),aa_set_bool_set_bool(insert_bool(X3),B3)))
    <=> ( ( pp(aa_set_bool_bool(member_bool(X3),B3))
         => pp(aa_set_bool_bool(ord_less_set_bool(A2),B3)) )
        & ( ~ pp(aa_set_bool_bool(member_bool(X3),B3))
         => ( ( pp(aa_set_bool_bool(member_bool(X3),A2))
             => pp(aa_set_bool_bool(ord_less_set_bool(minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))),B3)) )
            & ( ~ pp(aa_set_bool_bool(member_bool(X3),A2))
             => pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),B3)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_3703_psubset__insert__iff,axiom,
    ! [A2: set_nat,X3: nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),B3)))
    <=> ( ( pp(aa_set_nat_bool(member_nat(X3),B3))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A2),B3)) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X3),B3))
         => ( ( pp(aa_set_nat_bool(member_nat(X3),A2))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))),B3)) )
            & ( ~ pp(aa_set_nat_bool(member_nat(X3),A2))
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_3704_psubset__insert__iff,axiom,
    ! [A2: set_int,X3: int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),aa_set_int_set_int(insert_int(X3),B3)))
    <=> ( ( pp(aa_set_int_bool(member_int(X3),B3))
         => pp(aa_set_int_bool(ord_less_set_int(A2),B3)) )
        & ( ~ pp(aa_set_int_bool(member_int(X3),B3))
         => ( ( pp(aa_set_int_bool(member_int(X3),A2))
             => pp(aa_set_int_bool(ord_less_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int))),B3)) )
            & ( ~ pp(aa_set_int_bool(member_int(X3),A2))
             => pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_3705_set__replicate__Suc,axiom,
    ! [N: nat,X3: vEBT_VEBT] : aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(suc,N)),X3)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_3706_set__replicate__Suc,axiom,
    ! [N: nat,X3: real] : aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(replicate_real(aa_nat_nat(suc,N)),X3)) = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_3707_set__replicate__Suc,axiom,
    ! [N: nat,X3: bool] : aa_lis3587183946281582166t_bool(set_bool2,replicate_bool(aa_nat_nat(suc,N),X3)) = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) ).

% set_replicate_Suc
tff(fact_3708_set__replicate__Suc,axiom,
    ! [N: nat,X3: nat] : aa_list_nat_set_nat(set_nat2,replicate_nat(aa_nat_nat(suc,N),X3)) = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_3709_set__replicate__Suc,axiom,
    ! [N: nat,X3: int] : aa_list_int_set_int(set_int2,replicate_int(aa_nat_nat(suc,N),X3)) = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_3710_set__replicate__conv__if,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) ) ) ) ).

% set_replicate_conv_if
tff(fact_3711_set__replicate__conv__if,axiom,
    ! [N: nat,X3: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(replicate_real(N),X3)) = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(replicate_real(N),X3)) = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) ) ) ) ).

% set_replicate_conv_if
tff(fact_3712_set__replicate__conv__if,axiom,
    ! [N: nat,X3: bool] :
      ( ( ( N = zero_zero_nat )
       => ( aa_lis3587183946281582166t_bool(set_bool2,replicate_bool(N,X3)) = bot_bot_set_bool ) )
      & ( ( N != zero_zero_nat )
       => ( aa_lis3587183946281582166t_bool(set_bool2,replicate_bool(N,X3)) = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) ) ) ) ).

% set_replicate_conv_if
tff(fact_3713_set__replicate__conv__if,axiom,
    ! [N: nat,X3: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_list_nat_set_nat(set_nat2,replicate_nat(N,X3)) = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_list_nat_set_nat(set_nat2,replicate_nat(N,X3)) = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_3714_set__replicate__conv__if,axiom,
    ! [N: nat,X3: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_list_int_set_int(set_int2,replicate_int(N,X3)) = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_list_int_set_int(set_int2,replicate_int(N,X3)) = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ) ) ) ).

% set_replicate_conv_if
tff(fact_3715_prod__decode__aux_Opelims,axiom,
    ! [X3: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X3,Xa) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa)))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X3))
               => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Xa)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X3))
               => ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X3))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_3716_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE1225457145571040039t_assn,Xsi: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3717_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE2561185598974572172i_assn,Xsi: list_VEBT_VEBTi] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3718_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5577048379444288083T_assn,Xsi: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3719_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5641251438918192003t_assn,Xsi: list_int] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2806540629473551875Ti_int(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3720_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3721_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3722_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE123837653852366616t_assn,Xsi: list_int] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB392321844297299145t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3723_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re574865750389657758t_assn,Xsi: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L234762979517870878al_nat(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3724_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re560840401387337987i_assn,Xsi: list_VEBT_VEBTi] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3725_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re8050108611700907676T_assn,Xsi: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_3726_the__elem__eq,axiom,
    ! [X3: vEBT_VEBT] : the_elem_VEBT_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)) = X3 ).

% the_elem_eq
tff(fact_3727_the__elem__eq,axiom,
    ! [X3: real] : the_elem_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) = X3 ).

% the_elem_eq
tff(fact_3728_the__elem__eq,axiom,
    ! [X3: bool] :
      ( pp(aa_set_bool_bool(the_elem_bool,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)))
    <=> pp(X3) ) ).

% the_elem_eq
tff(fact_3729_the__elem__eq,axiom,
    ! [X3: nat] : aa_set_nat_nat(the_elem_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) = X3 ).

% the_elem_eq
tff(fact_3730_the__elem__eq,axiom,
    ! [X3: int] : aa_set_int_int(the_elem_int,aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) = X3 ).

% the_elem_eq
tff(fact_3731_sum__diff1_H__aux,axiom,
    ! [F4: set_VEBT_VEBT,I5: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_ep(I5),F))),F4))
       => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
           => ( groups3521240112447731263omplex(F,minus_5127226145743854075T_VEBT(I5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups3521240112447731263omplex(F,I5)),aa_VEBT_VEBT_complex(F,I)) ) )
          & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
           => ( groups3521240112447731263omplex(F,minus_5127226145743854075T_VEBT(I5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))) = groups3521240112447731263omplex(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3732_sum__diff1_H__aux,axiom,
    ! [F4: set_complex,I5: set_complex,F: fun_complex_complex,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ex(I5),F))),F4))
       => ( ( pp(aa_set_complex_bool(member_complex(I),I5))
           => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I5,insert_complex(I,bot_bot_set_complex))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups808145749697022017omplex(F,I5)),aa_complex_complex(F,I)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(I),I5))
           => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I5,insert_complex(I,bot_bot_set_complex))) = groups808145749697022017omplex(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3733_sum__diff1_H__aux,axiom,
    ! [F4: set_Code_integer,I5: set_Code_integer,F: fun_Co1900716531650368264omplex,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F4))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(collect_Code_integer(aa_fun5910925830913328781r_bool(aTP_Lamm_ez(I5),F))),F4))
       => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
           => ( groups3262226078671967728omplex(F,minus_2355218937544613996nteger(I5,insert_Code_integer(I,bot_bo3990330152332043303nteger))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups3262226078671967728omplex(F,I5)),aa_Cod5621350732091595593omplex(F,I)) ) )
          & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
           => ( groups3262226078671967728omplex(F,minus_2355218937544613996nteger(I5,insert_Code_integer(I,bot_bo3990330152332043303nteger))) = groups3262226078671967728omplex(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3734_sum__diff1_H__aux,axiom,
    ! [F4: set_real,I5: set_real,F: fun_real_complex,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,F4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_er(I5),F))),F4))
       => ( ( pp(aa_set_real_bool(member_real(I),I5))
           => ( groups5683813829254066239omplex(F,minus_minus_set_real(I5,aa_set_real_set_real(insert_real(I),bot_bot_set_real))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups5683813829254066239omplex(F,I5)),aa_real_complex(F,I)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(I),I5))
           => ( groups5683813829254066239omplex(F,minus_minus_set_real(I5,aa_set_real_set_real(insert_real(I),bot_bot_set_real))) = groups5683813829254066239omplex(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3735_sum__diff1_H__aux,axiom,
    ! [F4: set_bool,I5: set_bool,F: fun_bool_complex,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F4))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(collect_bool(aa_fun4000492451996535199l_bool(aTP_Lamm_le(I5),F))),F4))
       => ( ( pp(aa_set_bool_bool(member_bool(I),I5))
           => ( groups9190275049671490082omplex(F,minus_minus_set_bool(I5,aa_set_bool_set_bool(insert_bool(I),bot_bot_set_bool))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups9190275049671490082omplex(F,I5)),aa_bool_complex(F,I)) ) )
          & ( ~ pp(aa_set_bool_bool(member_bool(I),I5))
           => ( groups9190275049671490082omplex(F,minus_minus_set_bool(I5,aa_set_bool_set_bool(insert_bool(I),bot_bot_set_bool))) = groups9190275049671490082omplex(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3736_sum__diff1_H__aux,axiom,
    ! [F4: set_VEBT_VEBT,I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,F4))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_fb(I5),F))),F4))
       => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
           => ( groups7811786883911161277T_real(F,minus_5127226145743854075T_VEBT(I5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))) = aa_real_real(minus_minus_real(groups7811786883911161277T_real(F,I5)),aa_VEBT_VEBT_real(F,I)) ) )
          & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
           => ( groups7811786883911161277T_real(F,minus_5127226145743854075T_VEBT(I5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))) = groups7811786883911161277T_real(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3737_sum__diff1_H__aux,axiom,
    ! [F4: set_complex,I5: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_lf(I5),F))),F4))
       => ( ( pp(aa_set_complex_bool(member_complex(I),I5))
           => ( groups5737402329758386879x_real(F,minus_811609699411566653omplex(I5,insert_complex(I,bot_bot_set_complex))) = aa_real_real(minus_minus_real(groups5737402329758386879x_real(F,I5)),aa_complex_real(F,I)) ) )
          & ( ~ pp(aa_set_complex_bool(member_complex(I),I5))
           => ( groups5737402329758386879x_real(F,minus_811609699411566653omplex(I5,insert_complex(I,bot_bot_set_complex))) = groups5737402329758386879x_real(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3738_sum__diff1_H__aux,axiom,
    ! [F4: set_Code_integer,I5: set_Code_integer,F: fun_Co3960896333048504454r_real,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F4))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(collect_Code_integer(aa_fun7028086442024493327r_bool(aTP_Lamm_lg(I5),F))),F4))
       => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
           => ( groups9040905619451072238r_real(F,minus_2355218937544613996nteger(I5,insert_Code_integer(I,bot_bo3990330152332043303nteger))) = aa_real_real(minus_minus_real(groups9040905619451072238r_real(F,I5)),aa_Code_integer_real(F,I)) ) )
          & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
           => ( groups9040905619451072238r_real(F,minus_2355218937544613996nteger(I5,insert_Code_integer(I,bot_bo3990330152332043303nteger))) = groups9040905619451072238r_real(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3739_sum__diff1_H__aux,axiom,
    ! [F4: set_real,I5: set_real,F: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,F4))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fd(I5),F))),F4))
       => ( ( pp(aa_set_real_bool(member_real(I),I5))
           => ( groups97945582718554045l_real(F,minus_minus_set_real(I5,aa_set_real_set_real(insert_real(I),bot_bot_set_real))) = aa_real_real(minus_minus_real(groups97945582718554045l_real(F,I5)),aa_real_real(F,I)) ) )
          & ( ~ pp(aa_set_real_bool(member_real(I),I5))
           => ( groups97945582718554045l_real(F,minus_minus_set_real(I5,aa_set_real_set_real(insert_real(I),bot_bot_set_real))) = groups97945582718554045l_real(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3740_sum__diff1_H__aux,axiom,
    ! [F4: set_bool,I5: set_bool,F: fun_bool_real,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,F4))
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(collect_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_lh(I5),F))),F4))
       => ( ( pp(aa_set_bool_bool(member_bool(I),I5))
           => ( groups4964499404318219808l_real(F,minus_minus_set_bool(I5,aa_set_bool_set_bool(insert_bool(I),bot_bot_set_bool))) = aa_real_real(minus_minus_real(groups4964499404318219808l_real(F,I5)),aa_bool_real(F,I)) ) )
          & ( ~ pp(aa_set_bool_bool(member_bool(I),I5))
           => ( groups4964499404318219808l_real(F,minus_minus_set_bool(I5,aa_set_bool_set_bool(insert_bool(I),bot_bot_set_bool))) = groups4964499404318219808l_real(F,I5) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3741_is__singletonI,axiom,
    ! [X3: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(is_sin24926331636114728T_VEBT,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) ).

% is_singletonI
tff(fact_3742_is__singletonI,axiom,
    ! [X3: real] : pp(aa_set_real_bool(is_singleton_real,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) ).

% is_singletonI
tff(fact_3743_is__singletonI,axiom,
    ! [X3: bool] : pp(aa_set_bool_bool(is_singleton_bool,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) ).

% is_singletonI
tff(fact_3744_is__singletonI,axiom,
    ! [X3: nat] : pp(aa_set_nat_bool(is_singleton_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) ).

% is_singletonI
tff(fact_3745_is__singletonI,axiom,
    ! [X3: int] : pp(aa_set_int_bool(is_singleton_int,aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) ).

% is_singletonI
tff(fact_3746_sum__diff1_H,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_ep(I5),F))))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups3521240112447731263omplex(F,minus_5127226145743854075T_VEBT(I5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups3521240112447731263omplex(F,I5)),aa_VEBT_VEBT_complex(F,I)) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups3521240112447731263omplex(F,minus_5127226145743854075T_VEBT(I5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))) = groups3521240112447731263omplex(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3747_sum__diff1_H,axiom,
    ! [I5: set_complex,F: fun_complex_complex,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ex(I5),F))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I5,insert_complex(I,bot_bot_set_complex))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups808145749697022017omplex(F,I5)),aa_complex_complex(F,I)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I5))
         => ( groups808145749697022017omplex(F,minus_811609699411566653omplex(I5,insert_complex(I,bot_bot_set_complex))) = groups808145749697022017omplex(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3748_sum__diff1_H,axiom,
    ! [I5: set_Code_integer,F: fun_Co1900716531650368264omplex,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aTP_Lamm_ez(I5),F))))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
         => ( groups3262226078671967728omplex(F,minus_2355218937544613996nteger(I5,insert_Code_integer(I,bot_bo3990330152332043303nteger))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups3262226078671967728omplex(F,I5)),aa_Cod5621350732091595593omplex(F,I)) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
         => ( groups3262226078671967728omplex(F,minus_2355218937544613996nteger(I5,insert_Code_integer(I,bot_bo3990330152332043303nteger))) = groups3262226078671967728omplex(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3749_sum__diff1_H,axiom,
    ! [I5: set_real,F: fun_real_complex,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_er(I5),F))))
     => ( ( pp(aa_set_real_bool(member_real(I),I5))
         => ( groups5683813829254066239omplex(F,minus_minus_set_real(I5,aa_set_real_set_real(insert_real(I),bot_bot_set_real))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups5683813829254066239omplex(F,I5)),aa_real_complex(F,I)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I5))
         => ( groups5683813829254066239omplex(F,minus_minus_set_real(I5,aa_set_real_set_real(insert_real(I),bot_bot_set_real))) = groups5683813829254066239omplex(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3750_sum__diff1_H,axiom,
    ! [I5: set_bool,F: fun_bool_complex,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun4000492451996535199l_bool(aTP_Lamm_le(I5),F))))
     => ( ( pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups9190275049671490082omplex(F,minus_minus_set_bool(I5,aa_set_bool_set_bool(insert_bool(I),bot_bot_set_bool))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups9190275049671490082omplex(F,I5)),aa_bool_complex(F,I)) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups9190275049671490082omplex(F,minus_minus_set_bool(I5,aa_set_bool_set_bool(insert_bool(I),bot_bot_set_bool))) = groups9190275049671490082omplex(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3751_sum__diff1_H,axiom,
    ! [I5: set_int,F: fun_int_complex,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_ev(I5),F))))
     => ( ( pp(aa_set_int_bool(member_int(I),I5))
         => ( groups267424677133301183omplex(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,I5),aa_set_int_set_int(insert_int(I),bot_bot_set_int))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups267424677133301183omplex(F,I5)),aa_int_complex(F,I)) ) )
        & ( ~ pp(aa_set_int_bool(member_int(I),I5))
         => ( groups267424677133301183omplex(F,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,I5),aa_set_int_set_int(insert_int(I),bot_bot_set_int))) = groups267424677133301183omplex(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3752_sum__diff1_H,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_fb(I5),F))))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups7811786883911161277T_real(F,minus_5127226145743854075T_VEBT(I5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))) = aa_real_real(minus_minus_real(groups7811786883911161277T_real(F,I5)),aa_VEBT_VEBT_real(F,I)) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups7811786883911161277T_real(F,minus_5127226145743854075T_VEBT(I5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))) = groups7811786883911161277T_real(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3753_sum__diff1_H,axiom,
    ! [I5: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_lf(I5),F))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( groups5737402329758386879x_real(F,minus_811609699411566653omplex(I5,insert_complex(I,bot_bot_set_complex))) = aa_real_real(minus_minus_real(groups5737402329758386879x_real(F,I5)),aa_complex_real(F,I)) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I5))
         => ( groups5737402329758386879x_real(F,minus_811609699411566653omplex(I5,insert_complex(I,bot_bot_set_complex))) = groups5737402329758386879x_real(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3754_sum__diff1_H,axiom,
    ! [I5: set_Code_integer,F: fun_Co3960896333048504454r_real,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun7028086442024493327r_bool(aTP_Lamm_lg(I5),F))))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
         => ( groups9040905619451072238r_real(F,minus_2355218937544613996nteger(I5,insert_Code_integer(I,bot_bo3990330152332043303nteger))) = aa_real_real(minus_minus_real(groups9040905619451072238r_real(F,I5)),aa_Code_integer_real(F,I)) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
         => ( groups9040905619451072238r_real(F,minus_2355218937544613996nteger(I5,insert_Code_integer(I,bot_bo3990330152332043303nteger))) = groups9040905619451072238r_real(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3755_sum__diff1_H,axiom,
    ! [I5: set_real,F: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fd(I5),F))))
     => ( ( pp(aa_set_real_bool(member_real(I),I5))
         => ( groups97945582718554045l_real(F,minus_minus_set_real(I5,aa_set_real_set_real(insert_real(I),bot_bot_set_real))) = aa_real_real(minus_minus_real(groups97945582718554045l_real(F,I5)),aa_real_real(F,I)) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I5))
         => ( groups97945582718554045l_real(F,minus_minus_set_real(I5,aa_set_real_set_real(insert_real(I),bot_bot_set_real))) = groups97945582718554045l_real(F,I5) ) ) ) ) ).

% sum_diff1'
tff(fact_3756_set__removeAll,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] : aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,removeAll_VEBT_VEBT(X3,Xs)) = minus_5127226145743854075T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)) ).

% set_removeAll
tff(fact_3757_set__removeAll,axiom,
    ! [X3: real,Xs: list_real] : aa_lis3769085826154824284t_real(set_real2,removeAll_real(X3,Xs)) = minus_minus_set_real(aa_lis3769085826154824284t_real(set_real2,Xs),aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) ).

% set_removeAll
tff(fact_3758_set__removeAll,axiom,
    ! [X3: bool,Xs: list_bool] : aa_lis3587183946281582166t_bool(set_bool2,removeAll_bool(X3,Xs)) = minus_minus_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs),aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)) ).

% set_removeAll
tff(fact_3759_set__removeAll,axiom,
    ! [X3: int,Xs: list_int] : aa_list_int_set_int(set_int2,removeAll_int(X3,Xs)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_list_int_set_int(set_int2,Xs)),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) ).

% set_removeAll
tff(fact_3760_set__removeAll,axiom,
    ! [X3: nat,Xs: list_nat] : aa_list_nat_set_nat(set_nat2,removeAll_nat(X3,Xs)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_list_nat_set_nat(set_nat2,Xs)),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) ).

% set_removeAll
tff(fact_3761_prod_Oinsert_H,axiom,
    ! [I5: set_bool,P3: fun_bool_real,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_li(I5),P3))))
     => ( ( pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups4751087096006904181l_real(P3,aa_set_bool_set_bool(insert_bool(I),I5)) = groups4751087096006904181l_real(P3,I5) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups4751087096006904181l_real(P3,aa_set_bool_set_bool(insert_bool(I),I5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_bool_real(P3,I)),groups4751087096006904181l_real(P3,I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3762_prod_Oinsert_H,axiom,
    ! [I5: set_VEBT_VEBT,P3: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_du(I5),P3))))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups5615669421813869842T_real(P3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),I5)) = groups5615669421813869842T_real(P3,I5) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups5615669421813869842T_real(P3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),I5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(P3,I)),groups5615669421813869842T_real(P3,I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3763_prod_Oinsert_H,axiom,
    ! [I5: set_real,P3: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dw(I5),P3))))
     => ( ( pp(aa_set_real_bool(member_real(I),I5))
         => ( groups1542684614274415016l_real(P3,aa_set_real_set_real(insert_real(I),I5)) = groups1542684614274415016l_real(P3,I5) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I5))
         => ( groups1542684614274415016l_real(P3,aa_set_real_set_real(insert_real(I),I5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(P3,I)),groups1542684614274415016l_real(P3,I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3764_prod_Oinsert_H,axiom,
    ! [I5: set_nat,P3: fun_nat_real,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dy(I5),P3))))
     => ( ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( aa_set_nat_real(groups2144956460914285644t_real(P3),aa_set_nat_set_nat(insert_nat(I),I5)) = aa_set_nat_real(groups2144956460914285644t_real(P3),I5) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
         => ( aa_set_nat_real(groups2144956460914285644t_real(P3),aa_set_nat_set_nat(insert_nat(I),I5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(P3,I)),aa_set_nat_real(groups2144956460914285644t_real(P3),I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3765_prod_Oinsert_H,axiom,
    ! [I5: set_int,P3: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ea(I5),P3))))
     => ( ( pp(aa_set_int_bool(member_int(I),I5))
         => ( groups4331878035607307432t_real(P3,aa_set_int_set_int(insert_int(I),I5)) = groups4331878035607307432t_real(P3,I5) ) )
        & ( ~ pp(aa_set_int_bool(member_int(I),I5))
         => ( groups4331878035607307432t_real(P3,aa_set_int_set_int(insert_int(I),I5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(P3,I)),groups4331878035607307432t_real(P3,I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3766_prod_Oinsert_H,axiom,
    ! [I5: set_complex,P3: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ec(I5),P3))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( groups5446087311781667882x_real(P3,insert_complex(I,I5)) = groups5446087311781667882x_real(P3,I5) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I5))
         => ( groups5446087311781667882x_real(P3,insert_complex(I,I5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(P3,I)),groups5446087311781667882x_real(P3,I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3767_prod_Oinsert_H,axiom,
    ! [I5: set_Code_integer,P3: fun_Co3960896333048504454r_real,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun7028086442024493327r_bool(aTP_Lamm_ee(I5),P3))))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
         => ( groups7484263781351690841r_real(P3,insert_Code_integer(I,I5)) = groups7484263781351690841r_real(P3,I5) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
         => ( groups7484263781351690841r_real(P3,insert_Code_integer(I,I5)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(P3,I)),groups7484263781351690841r_real(P3,I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3768_prod_Oinsert_H,axiom,
    ! [I5: set_bool,P3: fun_bool_rat,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun4510340986217973961l_bool(aTP_Lamm_lj(I5),P3))))
     => ( ( pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups2627207557660832033ol_rat(P3,aa_set_bool_set_bool(insert_bool(I),I5)) = groups2627207557660832033ol_rat(P3,I5) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups2627207557660832033ol_rat(P3,aa_set_bool_set_bool(insert_bool(I),I5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_bool_rat(P3,I)),groups2627207557660832033ol_rat(P3,I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3769_prod_Oinsert_H,axiom,
    ! [I5: set_VEBT_VEBT,P3: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1017860439744518025T_bool(aTP_Lamm_eg(I5),P3))))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups7596523497119011646BT_rat(P3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),I5)) = groups7596523497119011646BT_rat(P3,I5) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups7596523497119011646BT_rat(P3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),I5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_VEBT_VEBT_rat(P3,I)),groups7596523497119011646BT_rat(P3,I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3770_prod_Oinsert_H,axiom,
    ! [I5: set_real,P3: fun_real_rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_ei(I5),P3))))
     => ( ( pp(aa_set_real_bool(member_real(I),I5))
         => ( groups6077134973956688596al_rat(P3,aa_set_real_set_real(insert_real(I),I5)) = groups6077134973956688596al_rat(P3,I5) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I5))
         => ( groups6077134973956688596al_rat(P3,aa_set_real_set_real(insert_real(I),I5)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(P3,I)),groups6077134973956688596al_rat(P3,I5)) ) ) ) ) ).

% prod.insert'
tff(fact_3771_sum_Oempty_H,axiom,
    ! [P3: fun_real_complex] : groups5683813829254066239omplex(P3,bot_bot_set_real) = zero_zero_complex ).

% sum.empty'
tff(fact_3772_sum_Oempty_H,axiom,
    ! [P3: fun_real_real] : groups97945582718554045l_real(P3,bot_bot_set_real) = zero_zero_real ).

% sum.empty'
tff(fact_3773_sum_Oempty_H,axiom,
    ! [P3: fun_real_rat] : groups3269169158384524137al_rat(P3,bot_bot_set_real) = zero_zero_rat ).

% sum.empty'
tff(fact_3774_sum_Oempty_H,axiom,
    ! [P3: fun_real_nat] : groups3904299218471019873al_nat(P3,bot_bot_set_real) = zero_zero_nat ).

% sum.empty'
tff(fact_3775_sum_Oempty_H,axiom,
    ! [P3: fun_real_int] : groups3901808747961969597al_int(P3,bot_bot_set_real) = zero_zero_int ).

% sum.empty'
tff(fact_3776_sum_Oempty_H,axiom,
    ! [P3: fun_bool_complex] : groups9190275049671490082omplex(P3,bot_bot_set_bool) = zero_zero_complex ).

% sum.empty'
tff(fact_3777_sum_Oempty_H,axiom,
    ! [P3: fun_bool_real] : groups4964499404318219808l_real(P3,bot_bot_set_bool) = zero_zero_real ).

% sum.empty'
tff(fact_3778_sum_Oempty_H,axiom,
    ! [P3: fun_bool_rat] : groups689945893769603916ol_rat(P3,bot_bot_set_bool) = zero_zero_rat ).

% sum.empty'
tff(fact_3779_sum_Oempty_H,axiom,
    ! [P3: fun_bool_nat] : groups1325075953856099652ol_nat(P3,bot_bot_set_bool) = zero_zero_nat ).

% sum.empty'
tff(fact_3780_sum_Oempty_H,axiom,
    ! [P3: fun_bool_int] : groups1322585483347049376ol_int(P3,bot_bot_set_bool) = zero_zero_int ).

% sum.empty'
tff(fact_3781_prod_Oempty_H,axiom,
    ! [P3: fun_real_assn] : groups5834761644390034980l_assn(P3,bot_bot_set_real) = one_one_assn ).

% prod.empty'
tff(fact_3782_prod_Oempty_H,axiom,
    ! [P3: fun_real_real] : groups1542684614274415016l_real(P3,bot_bot_set_real) = one_one_real ).

% prod.empty'
tff(fact_3783_prod_Oempty_H,axiom,
    ! [P3: fun_real_rat] : groups6077134973956688596al_rat(P3,bot_bot_set_real) = one_one_rat ).

% prod.empty'
tff(fact_3784_prod_Oempty_H,axiom,
    ! [P3: fun_real_nat] : groups6712265034043184332al_nat(P3,bot_bot_set_real) = one_one_nat ).

% prod.empty'
tff(fact_3785_prod_Oempty_H,axiom,
    ! [P3: fun_real_int] : groups6709774563534134056al_int(P3,bot_bot_set_real) = one_one_int ).

% prod.empty'
tff(fact_3786_prod_Oempty_H,axiom,
    ! [P3: fun_bool_assn] : groups9008821520179477105l_assn(P3,bot_bot_set_bool) = one_one_assn ).

% prod.empty'
tff(fact_3787_prod_Oempty_H,axiom,
    ! [P3: fun_bool_real] : groups4751087096006904181l_real(P3,bot_bot_set_bool) = one_one_real ).

% prod.empty'
tff(fact_3788_prod_Oempty_H,axiom,
    ! [P3: fun_bool_rat] : groups2627207557660832033ol_rat(P3,bot_bot_set_bool) = one_one_rat ).

% prod.empty'
tff(fact_3789_prod_Oempty_H,axiom,
    ! [P3: fun_bool_nat] : groups3262337617747327769ol_nat(P3,bot_bot_set_bool) = one_one_nat ).

% prod.empty'
tff(fact_3790_prod_Oempty_H,axiom,
    ! [P3: fun_bool_int] : groups3259847147238277493ol_int(P3,bot_bot_set_bool) = one_one_int ).

% prod.empty'
tff(fact_3791_listI__assn__finite,axiom,
    ! [I5: set_nat,A2: fun_VE905575822219544865i_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( vEBT_L1528199826722428489_VEBTi(I5,A2,Xs,Xsi) = bot_bot_assn ) ) ).

% listI_assn_finite
tff(fact_3792_sum_Oinsert_H,axiom,
    ! [I5: set_bool,P3: fun_bool_complex,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun4000492451996535199l_bool(aTP_Lamm_le(I5),P3))))
     => ( ( pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups9190275049671490082omplex(P3,aa_set_bool_set_bool(insert_bool(I),I5)) = groups9190275049671490082omplex(P3,I5) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups9190275049671490082omplex(P3,aa_set_bool_set_bool(insert_bool(I),I5)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_bool_complex(P3,I)),groups9190275049671490082omplex(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3793_sum_Oinsert_H,axiom,
    ! [I5: set_VEBT_VEBT,P3: fun_VE5569070535936029429omplex,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_ep(I5),P3))))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups3521240112447731263omplex(P3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),I5)) = groups3521240112447731263omplex(P3,I5) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups3521240112447731263omplex(P3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),I5)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_VEBT_VEBT_complex(P3,I)),groups3521240112447731263omplex(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3794_sum_Oinsert_H,axiom,
    ! [I5: set_real,P3: fun_real_complex,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_er(I5),P3))))
     => ( ( pp(aa_set_real_bool(member_real(I),I5))
         => ( groups5683813829254066239omplex(P3,aa_set_real_set_real(insert_real(I),I5)) = groups5683813829254066239omplex(P3,I5) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I5))
         => ( groups5683813829254066239omplex(P3,aa_set_real_set_real(insert_real(I),I5)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(P3,I)),groups5683813829254066239omplex(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3795_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P3: fun_nat_complex,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_et(I5),P3))))
     => ( ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( groups8515261248781899619omplex(P3,aa_set_nat_set_nat(insert_nat(I),I5)) = groups8515261248781899619omplex(P3,I5) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
         => ( groups8515261248781899619omplex(P3,aa_set_nat_set_nat(insert_nat(I),I5)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(P3,I)),groups8515261248781899619omplex(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3796_sum_Oinsert_H,axiom,
    ! [I5: set_int,P3: fun_int_complex,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_ev(I5),P3))))
     => ( ( pp(aa_set_int_bool(member_int(I),I5))
         => ( groups267424677133301183omplex(P3,aa_set_int_set_int(insert_int(I),I5)) = groups267424677133301183omplex(P3,I5) ) )
        & ( ~ pp(aa_set_int_bool(member_int(I),I5))
         => ( groups267424677133301183omplex(P3,aa_set_int_set_int(insert_int(I),I5)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_int_complex(P3,I)),groups267424677133301183omplex(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3797_sum_Oinsert_H,axiom,
    ! [I5: set_complex,P3: fun_complex_complex,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ex(I5),P3))))
     => ( ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( groups808145749697022017omplex(P3,insert_complex(I,I5)) = groups808145749697022017omplex(P3,I5) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(I),I5))
         => ( groups808145749697022017omplex(P3,insert_complex(I,I5)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(P3,I)),groups808145749697022017omplex(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3798_sum_Oinsert_H,axiom,
    ! [I5: set_Code_integer,P3: fun_Co1900716531650368264omplex,I: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aTP_Lamm_ez(I5),P3))))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
         => ( groups3262226078671967728omplex(P3,insert_Code_integer(I,I5)) = groups3262226078671967728omplex(P3,I5) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
         => ( groups3262226078671967728omplex(P3,insert_Code_integer(I,I5)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_Cod5621350732091595593omplex(P3,I)),groups3262226078671967728omplex(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3799_sum_Oinsert_H,axiom,
    ! [I5: set_bool,P3: fun_bool_real,I: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,collect_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_lh(I5),P3))))
     => ( ( pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups4964499404318219808l_real(P3,aa_set_bool_set_bool(insert_bool(I),I5)) = groups4964499404318219808l_real(P3,I5) ) )
        & ( ~ pp(aa_set_bool_bool(member_bool(I),I5))
         => ( groups4964499404318219808l_real(P3,aa_set_bool_set_bool(insert_bool(I),I5)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_bool_real(P3,I)),groups4964499404318219808l_real(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3800_sum_Oinsert_H,axiom,
    ! [I5: set_VEBT_VEBT,P3: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_fb(I5),P3))))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups7811786883911161277T_real(P3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),I5)) = groups7811786883911161277T_real(P3,I5) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( groups7811786883911161277T_real(P3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),I5)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(P3,I)),groups7811786883911161277T_real(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3801_sum_Oinsert_H,axiom,
    ! [I5: set_real,P3: fun_real_real,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fd(I5),P3))))
     => ( ( pp(aa_set_real_bool(member_real(I),I5))
         => ( groups97945582718554045l_real(P3,aa_set_real_set_real(insert_real(I),I5)) = groups97945582718554045l_real(P3,I5) ) )
        & ( ~ pp(aa_set_real_bool(member_real(I),I5))
         => ( groups97945582718554045l_real(P3,aa_set_real_set_real(insert_real(I),I5)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(P3,I)),groups97945582718554045l_real(P3,I5)) ) ) ) ) ).

% sum.insert'
tff(fact_3802_sum_Onon__neutral_H,axiom,
    ! [G: fun_VE5569070535936029429omplex,I5: set_VEBT_VEBT] : groups3521240112447731263omplex(G,collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_lk(G),I5))) = groups3521240112447731263omplex(G,I5) ).

% sum.non_neutral'
tff(fact_3803_sum_Onon__neutral_H,axiom,
    ! [G: fun_real_complex,I5: set_real] : groups5683813829254066239omplex(G,collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_ll(G),I5))) = groups5683813829254066239omplex(G,I5) ).

% sum.non_neutral'
tff(fact_3804_sum_Onon__neutral_H,axiom,
    ! [G: fun_complex_complex,I5: set_complex] : groups808145749697022017omplex(G,collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_lm(G),I5))) = groups808145749697022017omplex(G,I5) ).

% sum.non_neutral'
tff(fact_3805_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_complex,I5: set_nat] : groups8515261248781899619omplex(G,collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_ln(G),I5))) = groups8515261248781899619omplex(G,I5) ).

% sum.non_neutral'
tff(fact_3806_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_complex,I5: set_int] : groups267424677133301183omplex(G,collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_lo(G),I5))) = groups267424677133301183omplex(G,I5) ).

% sum.non_neutral'
tff(fact_3807_sum_Onon__neutral_H,axiom,
    ! [G: fun_VEBT_VEBT_real,I5: set_VEBT_VEBT] : groups7811786883911161277T_real(G,collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_lp(G),I5))) = groups7811786883911161277T_real(G,I5) ).

% sum.non_neutral'
tff(fact_3808_sum_Onon__neutral_H,axiom,
    ! [G: fun_real_real,I5: set_real] : groups97945582718554045l_real(G,collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_lq(G),I5))) = groups97945582718554045l_real(G,I5) ).

% sum.non_neutral'
tff(fact_3809_sum_Onon__neutral_H,axiom,
    ! [G: fun_complex_real,I5: set_complex] : groups5737402329758386879x_real(G,collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_lr(G),I5))) = groups5737402329758386879x_real(G,I5) ).

% sum.non_neutral'
tff(fact_3810_sum_Onon__neutral_H,axiom,
    ! [G: fun_nat_real,I5: set_nat] : aa_set_nat_real(groups8560362682196896993t_real(G),collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_ls(G),I5))) = aa_set_nat_real(groups8560362682196896993t_real(G),I5) ).

% sum.non_neutral'
tff(fact_3811_sum_Onon__neutral_H,axiom,
    ! [G: fun_int_real,I5: set_int] : groups1523912220035142973t_real(G,collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_lt(G),I5))) = groups1523912220035142973t_real(G,I5) ).

% sum.non_neutral'
tff(fact_3812_prod_Onon__neutral_H,axiom,
    ! [G: fun_VEBT_VEBT_assn,I5: set_VEBT_VEBT] : groups8339234656476930702T_assn(G,collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_lu(G),I5))) = groups8339234656476930702T_assn(G,I5) ).

% prod.non_neutral'
tff(fact_3813_prod_Onon__neutral_H,axiom,
    ! [G: fun_real_assn,I5: set_real] : groups5834761644390034980l_assn(G,collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_lv(G),I5))) = groups5834761644390034980l_assn(G,I5) ).

% prod.non_neutral'
tff(fact_3814_prod_Onon__neutral_H,axiom,
    ! [G: fun_complex_assn,I5: set_complex] : groups8985371454280824230x_assn(G,collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_lw(G),I5))) = groups8985371454280824230x_assn(G,I5) ).

% prod.non_neutral'
tff(fact_3815_prod_Onon__neutral_H,axiom,
    ! [G: fun_nat_assn,I5: set_nat] : groups3916077543535352520t_assn(G,collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_lx(G),I5))) = groups3916077543535352520t_assn(G,I5) ).

% prod.non_neutral'
tff(fact_3816_prod_Onon__neutral_H,axiom,
    ! [G: fun_int_assn,I5: set_int] : groups4891613008741529892t_assn(G,collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_ly(G),I5))) = groups4891613008741529892t_assn(G,I5) ).

% prod.non_neutral'
tff(fact_3817_prod_Onon__neutral_H,axiom,
    ! [G: fun_VEBT_VEBT_real,I5: set_VEBT_VEBT] : groups5615669421813869842T_real(G,collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_lz(G),I5))) = groups5615669421813869842T_real(G,I5) ).

% prod.non_neutral'
tff(fact_3818_prod_Onon__neutral_H,axiom,
    ! [G: fun_real_real,I5: set_real] : groups1542684614274415016l_real(G,collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_ma(G),I5))) = groups1542684614274415016l_real(G,I5) ).

% prod.non_neutral'
tff(fact_3819_prod_Onon__neutral_H,axiom,
    ! [G: fun_complex_real,I5: set_complex] : groups5446087311781667882x_real(G,collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_mb(G),I5))) = groups5446087311781667882x_real(G,I5) ).

% prod.non_neutral'
tff(fact_3820_prod_Onon__neutral_H,axiom,
    ! [G: fun_nat_real,I5: set_nat] : aa_set_nat_real(groups2144956460914285644t_real(G),collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_mc(G),I5))) = aa_set_nat_real(groups2144956460914285644t_real(G),I5) ).

% prod.non_neutral'
tff(fact_3821_prod_Onon__neutral_H,axiom,
    ! [G: fun_int_real,I5: set_int] : groups4331878035607307432t_real(G,collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_md(G),I5))) = groups4331878035607307432t_real(G,I5) ).

% prod.non_neutral'
tff(fact_3822_is__singleton__the__elem,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(is_sin24926331636114728T_VEBT,A2))
    <=> ( A2 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(the_elem_VEBT_VEBT(A2)),bot_bo8194388402131092736T_VEBT) ) ) ).

% is_singleton_the_elem
tff(fact_3823_is__singleton__the__elem,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(is_singleton_real,A2))
    <=> ( A2 = aa_set_real_set_real(insert_real(the_elem_real(A2)),bot_bot_set_real) ) ) ).

% is_singleton_the_elem
tff(fact_3824_is__singleton__the__elem,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(is_singleton_bool,A2))
    <=> ( A2 = aa_set_bool_set_bool(insert_bool(aa_set_bool_bool(the_elem_bool,A2)),bot_bot_set_bool) ) ) ).

% is_singleton_the_elem
tff(fact_3825_is__singleton__the__elem,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(is_singleton_nat,A2))
    <=> ( A2 = aa_set_nat_set_nat(insert_nat(aa_set_nat_nat(the_elem_nat,A2)),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_3826_is__singleton__the__elem,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(is_singleton_int,A2))
    <=> ( A2 = aa_set_int_set_int(insert_int(aa_set_int_int(the_elem_int,A2)),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_3827_is__singletonI_H,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( A2 != bot_bo8194388402131092736T_VEBT )
     => ( ! [X: vEBT_VEBT,Y3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y3),A2))
             => ( X = Y3 ) ) )
       => pp(aa_set8260762808118238015T_bool(is_sin24926331636114728T_VEBT,A2)) ) ) ).

% is_singletonI'
tff(fact_3828_is__singletonI_H,axiom,
    ! [A2: set_set_nat] :
      ( ( A2 != bot_bot_set_set_nat )
     => ( ! [X: set_nat,Y3: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
           => ( pp(aa_set_set_nat_bool(member_set_nat(Y3),A2))
             => ( X = Y3 ) ) )
       => pp(aa_set_set_nat_bool(is_singleton_set_nat,A2)) ) ) ).

% is_singletonI'
tff(fact_3829_is__singletonI_H,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
     => ( ! [X: real,Y3: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => ( pp(aa_set_real_bool(member_real(Y3),A2))
             => ( X = Y3 ) ) )
       => pp(aa_set_real_bool(is_singleton_real,A2)) ) ) ).

% is_singletonI'
tff(fact_3830_is__singletonI_H,axiom,
    ! [A2: set_bool] :
      ( ( A2 != bot_bot_set_bool )
     => ( ! [X: bool,Y3: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),A2))
           => ( pp(aa_set_bool_bool(member_bool(Y3),A2))
             => ( pp(X)
              <=> pp(Y3) ) ) )
       => pp(aa_set_bool_bool(is_singleton_bool,A2)) ) ) ).

% is_singletonI'
tff(fact_3831_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X: nat,Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => ( pp(aa_set_nat_bool(member_nat(Y3),A2))
             => ( X = Y3 ) ) )
       => pp(aa_set_nat_bool(is_singleton_nat,A2)) ) ) ).

% is_singletonI'
tff(fact_3832_is__singletonI_H,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [X: int,Y3: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => ( pp(aa_set_int_bool(member_int(Y3),A2))
             => ( X = Y3 ) ) )
       => pp(aa_set_int_bool(is_singleton_int,A2)) ) ) ).

% is_singletonI'
tff(fact_3833_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( aa_set_nat_real(groups8560362682196896993t_real(aa_fun1839206346889086768t_real(aTP_Lamm_me(G),H)),I5) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups8560362682196896993t_real(G),I5)),aa_set_nat_real(groups8560362682196896993t_real(H),I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3834_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( groups1523912220035142973t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mf(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups1523912220035142973t_real(G,I5)),groups1523912220035142973t_real(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3835_sum_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( groups5737402329758386879x_real(aa_fun8933909628229591020x_real(aTP_Lamm_mg(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5737402329758386879x_real(G,I5)),groups5737402329758386879x_real(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3836_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I5))
     => ( groups9040905619451072238r_real(aa_fun6341089502340155082r_real(aTP_Lamm_mh(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups9040905619451072238r_real(G,I5)),groups9040905619451072238r_real(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3837_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( groups1351286907653491341at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mi(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups1351286907653491341at_rat(G,I5)),groups1351286907653491341at_rat(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3838_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( groups2350640619554545897nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_mj(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2350640619554545897nt_rat(G,I5)),groups2350640619554545897nt_rat(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3839_sum_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( groups2276542476275365739ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_mk(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2276542476275365739ex_rat(G,I5)),groups2276542476275365739ex_rat(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3840_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: fun_Code_integer_rat,H: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I5))
     => ( groups8878813951405302554er_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_ml(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups8878813951405302554er_rat(G,I5)),groups8878813951405302554er_rat(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3841_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( aa_set_nat_nat(groups1986416967739987077at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_mm(G),H)),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups1986416967739987077at_nat(G),I5)),aa_set_nat_nat(groups1986416967739987077at_nat(H),I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3842_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( groups2985770679641041633nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_mn(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2985770679641041633nt_nat(G,I5)),groups2985770679641041633nt_nat(H,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_3843_length__removeAll__less__eq,axiom,
    ! [X3: real,Xs: list_real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,removeAll_real(X3,Xs))),aa_list_real_nat(size_size_list_real,Xs))) ).

% length_removeAll_less_eq
tff(fact_3844_length__removeAll__less__eq,axiom,
    ! [X3: bool,Xs: list_bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,removeAll_bool(X3,Xs))),aa_list_bool_nat(size_size_list_bool,Xs))) ).

% length_removeAll_less_eq
tff(fact_3845_length__removeAll__less__eq,axiom,
    ! [X3: nat,Xs: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,removeAll_nat(X3,Xs))),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% length_removeAll_less_eq
tff(fact_3846_length__removeAll__less__eq,axiom,
    ! [X3: int,Xs: list_int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,removeAll_int(X3,Xs))),aa_list_int_nat(size_size_list_int,Xs))) ).

% length_removeAll_less_eq
tff(fact_3847_prod_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( aa_set_nat_real(groups2144956460914285644t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mo(G),H)),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups2144956460914285644t_real(G),I5)),aa_set_nat_real(groups2144956460914285644t_real(H),I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3848_prod_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( groups4331878035607307432t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mp(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups4331878035607307432t_real(G,I5)),groups4331878035607307432t_real(H,I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3849_prod_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( groups5446087311781667882x_real(aa_fun8933909628229591020x_real(aTP_Lamm_mq(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups5446087311781667882x_real(G,I5)),groups5446087311781667882x_real(H,I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3850_prod_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I5))
     => ( groups7484263781351690841r_real(aa_fun6341089502340155082r_real(aTP_Lamm_mr(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups7484263781351690841r_real(G,I5)),groups7484263781351690841r_real(H,I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3851_prod_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( groups4112464933559648120at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ms(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups4112464933559648120at_rat(G,I5)),groups4112464933559648120at_rat(H,I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3852_prod_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( groups5111818645460702676nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_mt(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups5111818645460702676nt_rat(G,I5)),groups5111818645460702676nt_rat(H,I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3853_prod_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( groups6458467974770906710ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_mu(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups6458467974770906710ex_rat(G,I5)),groups6458467974770906710ex_rat(H,I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3854_prod_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: fun_Code_integer_rat,H: fun_Code_integer_rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I5))
     => ( groups7390404786021641221er_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_mv(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups7390404786021641221er_rat(G,I5)),groups7390404786021641221er_rat(H,I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3855_prod_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( aa_set_nat_nat(groups4747594993646143856at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_mw(G),H)),I5) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups4747594993646143856at_nat(G),I5)),aa_set_nat_nat(groups4747594993646143856at_nat(H),I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3856_prod_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( groups5746948705547198412nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_mx(G),H),I5) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups5746948705547198412nt_nat(G,I5)),groups5746948705547198412nt_nat(H,I5)) ) ) ).

% prod.distrib_triv'
tff(fact_3857_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_complex(G,X) = zero_zero_complex ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_complex(G,X) = aa_VEBT_VEBT_complex(H,X) ) )
         => ( groups3521240112447731263omplex(G,T3) = groups3521240112447731263omplex(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3858_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_complex(G,X) = zero_zero_complex ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_complex(G,X) = aa_real_complex(H,X) ) )
         => ( groups5683813829254066239omplex(G,T3) = groups5683813829254066239omplex(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3859_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_real(G,X) = zero_zero_real ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_real(G,X) = aa_VEBT_VEBT_real(H,X) ) )
         => ( groups7811786883911161277T_real(G,T3) = groups7811786883911161277T_real(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3860_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_real(G,X) = zero_zero_real ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
         => ( groups97945582718554045l_real(G,T3) = groups97945582718554045l_real(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3861_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_rat(G,X) = zero_zero_rat ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_rat(G,X) = aa_VEBT_VEBT_rat(H,X) ) )
         => ( groups7132981355262091625BT_rat(G,T3) = groups7132981355262091625BT_rat(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3862_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_rat(G,X) = zero_zero_rat ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_rat(G,X) = aa_real_rat(H,X) ) )
         => ( groups3269169158384524137al_rat(G,T3) = groups3269169158384524137al_rat(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3863_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,H: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_nat(G,X) = zero_zero_nat ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_nat(G,X) = aa_VEBT_VEBT_nat(H,X) ) )
         => ( groups7768111415348587361BT_nat(G,T3) = groups7768111415348587361BT_nat(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3864_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_nat(G,X) = zero_zero_nat ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_nat(G,X) = aa_real_nat(H,X) ) )
         => ( groups3904299218471019873al_nat(G,T3) = groups3904299218471019873al_nat(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3865_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,H: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_int(G,X) = zero_zero_int ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_int(G,X) = aa_VEBT_VEBT_int(H,X) ) )
         => ( groups7765620944839537085BT_int(G,T3) = groups7765620944839537085BT_int(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3866_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_int(G,X) = zero_zero_int ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_int(G,X) = aa_real_int(H,X) ) )
         => ( groups3901808747961969597al_int(G,T3) = groups3901808747961969597al_int(H,S3) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3867_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VE5569070535936029429omplex,G: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_complex(H,I2) = zero_zero_complex ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_complex(G,X) = aa_VEBT_VEBT_complex(H,X) ) )
         => ( groups3521240112447731263omplex(G,S3) = groups3521240112447731263omplex(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3868_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_complex(H,I2) = zero_zero_complex ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_complex(G,X) = aa_real_complex(H,X) ) )
         => ( groups5683813829254066239omplex(G,S3) = groups5683813829254066239omplex(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3869_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_real(H,I2) = zero_zero_real ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_real(G,X) = aa_VEBT_VEBT_real(H,X) ) )
         => ( groups7811786883911161277T_real(G,S3) = groups7811786883911161277T_real(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3870_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_real(H,I2) = zero_zero_real ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
         => ( groups97945582718554045l_real(G,S3) = groups97945582718554045l_real(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3871_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_rat(H,I2) = zero_zero_rat ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_rat(G,X) = aa_VEBT_VEBT_rat(H,X) ) )
         => ( groups7132981355262091625BT_rat(G,S3) = groups7132981355262091625BT_rat(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3872_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_rat(H,I2) = zero_zero_rat ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_rat(G,X) = aa_real_rat(H,X) ) )
         => ( groups3269169158384524137al_rat(G,S3) = groups3269169158384524137al_rat(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3873_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_nat(H,I2) = zero_zero_nat ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_nat(G,X) = aa_VEBT_VEBT_nat(H,X) ) )
         => ( groups7768111415348587361BT_nat(G,S3) = groups7768111415348587361BT_nat(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3874_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_nat(H,I2) = zero_zero_nat ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_nat(G,X) = aa_real_nat(H,X) ) )
         => ( groups3904299218471019873al_nat(G,S3) = groups3904299218471019873al_nat(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3875_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_int(H,I2) = zero_zero_int ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_int(G,X) = aa_VEBT_VEBT_int(H,X) ) )
         => ( groups7765620944839537085BT_int(G,S3) = groups7765620944839537085BT_int(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3876_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_int(H,I2) = zero_zero_int ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_int(G,X) = aa_real_int(H,X) ) )
         => ( groups3901808747961969597al_int(G,S3) = groups3901808747961969597al_int(H,T3) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3877_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_complex(G,X) = zero_zero_complex ) )
       => ( groups8515261248781899619omplex(G,T3) = groups8515261248781899619omplex(G,S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3878_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_real(G,X) = zero_zero_real ) )
       => ( aa_set_nat_real(groups8560362682196896993t_real(G),T3) = aa_set_nat_real(groups8560362682196896993t_real(G),S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3879_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_rat(G,X) = zero_zero_rat ) )
       => ( groups1351286907653491341at_rat(G,T3) = groups1351286907653491341at_rat(G,S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3880_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_nat(G,X) = zero_zero_nat ) )
       => ( aa_set_nat_nat(groups1986416967739987077at_nat(G),T3) = aa_set_nat_nat(groups1986416967739987077at_nat(G),S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3881_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_int(G,X) = zero_zero_int ) )
       => ( groups1983926497230936801at_int(G,T3) = groups1983926497230936801at_int(G,S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3882_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_complex(G,X) = zero_zero_complex ) )
       => ( groups267424677133301183omplex(G,T3) = groups267424677133301183omplex(G,S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3883_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_real(G,X) = zero_zero_real ) )
       => ( groups1523912220035142973t_real(G,T3) = groups1523912220035142973t_real(G,S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3884_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_rat(G,X) = zero_zero_rat ) )
       => ( groups2350640619554545897nt_rat(G,T3) = groups2350640619554545897nt_rat(G,S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3885_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_nat(G,X) = zero_zero_nat ) )
       => ( groups2985770679641041633nt_nat(G,T3) = groups2985770679641041633nt_nat(G,S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3886_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_int(G,X) = zero_zero_int ) )
       => ( aa_set_int_int(groups2983280209131991357nt_int(G),T3) = aa_set_int_int(groups2983280209131991357nt_int(G),S3) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3887_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_complex] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_complex(G,X) = zero_zero_complex ) )
       => ( groups8515261248781899619omplex(G,S3) = groups8515261248781899619omplex(G,T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3888_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_real(G,X) = zero_zero_real ) )
       => ( aa_set_nat_real(groups8560362682196896993t_real(G),S3) = aa_set_nat_real(groups8560362682196896993t_real(G),T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3889_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_rat(G,X) = zero_zero_rat ) )
       => ( groups1351286907653491341at_rat(G,S3) = groups1351286907653491341at_rat(G,T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3890_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_nat(G,X) = zero_zero_nat ) )
       => ( aa_set_nat_nat(groups1986416967739987077at_nat(G),S3) = aa_set_nat_nat(groups1986416967739987077at_nat(G),T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3891_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_int(G,X) = zero_zero_int ) )
       => ( groups1983926497230936801at_int(G,S3) = groups1983926497230936801at_int(G,T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3892_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_complex(G,X) = zero_zero_complex ) )
       => ( groups267424677133301183omplex(G,S3) = groups267424677133301183omplex(G,T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3893_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_real(G,X) = zero_zero_real ) )
       => ( groups1523912220035142973t_real(G,S3) = groups1523912220035142973t_real(G,T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3894_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_rat(G,X) = zero_zero_rat ) )
       => ( groups2350640619554545897nt_rat(G,S3) = groups2350640619554545897nt_rat(G,T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3895_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_nat(G,X) = zero_zero_nat ) )
       => ( groups2985770679641041633nt_nat(G,S3) = groups2985770679641041633nt_nat(G,T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3896_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_int(G,X) = zero_zero_int ) )
       => ( aa_set_int_int(groups2983280209131991357nt_int(G),S3) = aa_set_int_int(groups2983280209131991357nt_int(G),T3) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3897_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE2561185598974572172i_assn,A8: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,A2,Xs,Xsi) = vEBT_L886525131989349516_VEBTi(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3898_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE5577048379444288083T_assn,A8: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,A2,Xs,Xsi) = vEBT_L2497118539674116125T_VEBT(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3899_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE1571388752190420990T_assn,A8: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                        & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi) = vEBT_L3204528365124325536T_VEBT(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3900_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE1885522831824361987l_assn,A8: fun_VE1885522831824361987l_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( aa_list_real_nat(size_size_list_real,Xsi) = aa_list_real_nat(size_size_list_real,Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = aa_list_real_nat(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,A2,Xs,Xsi) = vEBT_L7728200936804140803i_real(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3901_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE6910866152645720856l_assn,A8: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( aa_list_real_nat(size_size_list_real,Xsi) = aa_list_real_nat(size_size_list_real,Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_real_nat(size_size_list_real,Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                        & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L4281036506115550016T_real(I5,A2,Xs,Xsi) = vEBT_L4281036506115550016T_real(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3902_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE3808733824349961398l_assn,A8: fun_VE3808733824349961398l_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_bool,Xsi2: list_bool] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( aa_list_bool_nat(size_size_list_bool,Xsi) = aa_list_bool_nat(size_size_list_bool,Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = aa_list_bool_nat(size_size_list_bool,Xsi) )
                     => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                        & ( pp(aa_nat_bool(nth_bool(Xsi),I2))
                        <=> pp(aa_nat_bool(nth_bool(Xsi2),I2)) ) ) ) ) )
             => ( vEBT_L2302322349616425472i_bool(I5,A2,Xs,Xsi) = vEBT_L2302322349616425472i_bool(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3903_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE1323945509527881313l_assn,A8: fun_VE1323945509527881313l_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_bool,Xsi2: list_bool] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( aa_list_bool_nat(size_size_list_bool,Xsi) = aa_list_bool_nat(size_size_list_bool,Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_bool_nat(size_size_list_bool,Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                        & ( pp(aa_nat_bool(nth_bool(Xsi),I2))
                        <=> pp(aa_nat_bool(nth_bool(Xsi2),I2)) ) ) ) ) )
             => ( vEBT_L6176004506706552451T_bool(I5,A2,Xs,Xsi) = vEBT_L6176004506706552451T_bool(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3904_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE1225457145571040039t_assn,A8: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( aa_list_nat_nat(size_size_list_nat,Xsi) = aa_list_nat_nat(size_size_list_nat,Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = aa_list_nat_nat(size_size_list_nat,Xsi) )
                     => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                        & ( aa_nat_nat(nth_nat(Xsi),I2) = aa_nat_nat(nth_nat(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi) = vEBT_L2809031099982602151Ti_nat(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3905_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE4931415397359990460t_assn,A8: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( aa_list_nat_nat(size_size_list_nat,Xsi) = aa_list_nat_nat(size_size_list_nat,Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                        & ( aa_nat_nat(nth_nat(Xsi),I2) = aa_nat_nat(nth_nat(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi) = vEBT_L8650695023172932196BT_nat(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3906_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE5641251438918192003t_assn,A8: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_int,Xsi2: list_int] :
      ( ( I5 = I6 )
     => ( ( A2 = A8 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( aa_list_int_nat(size_size_list_int,Xsi) = aa_list_int_nat(size_size_list_int,Xsi2) )
           => ( ! [I2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I2),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = aa_list_int_nat(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,A2,Xs,Xsi) = vEBT_L2806540629473551875Ti_int(I6,A8,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_3907_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,A2: fun_VE2561185598974572172i_assn,A8: fun_VE2561185598974572172i_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(A2,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(A8,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi) = vEBT_L886525131989349516_VEBTi(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3908_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,A2: fun_VE5577048379444288083T_assn,A8: fun_VE5577048379444288083T_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(A2,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(A8,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi) = vEBT_L2497118539674116125T_VEBT(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3909_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,A8: fun_VE1571388752190420990T_assn] :
      ( ( I5 = I6 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(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_VEB115603943081001463T_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2)) = aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A8,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi) = vEBT_L3204528365124325536T_VEBT(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3910_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real,A2: fun_VE1885522831824361987l_assn,A8: fun_VE1885522831824361987l_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( aa_list_real_nat(size_size_list_real,Xsi) = aa_list_real_nat(size_size_list_real,Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = aa_list_real_nat(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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)),aa_nat_real(nth_real(Xsi),I2)) = aa_real_assn(aa_VEB9107798209145535940l_assn(A8,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_real(nth_real(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L7728200936804140803i_real(I5,A2,Xs,Xsi) = vEBT_L7728200936804140803i_real(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3911_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A2: fun_VE6910866152645720856l_assn,A8: fun_VE6910866152645720856l_assn] :
      ( ( I5 = I6 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( aa_list_real_nat(size_size_list_real,Xsi) = aa_list_real_nat(size_size_list_real,Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_real_nat(size_size_list_real,Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                      & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) )
                      & ( aa_real_assn(aa_VEB8575627266564790985l_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)),aa_nat_real(nth_real(Xsi),I2)) = aa_real_assn(aa_VEB8575627266564790985l_assn(A8,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2)),aa_nat_real(nth_real(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L4281036506115550016T_real(I5,A2,Xs,Xsi) = vEBT_L4281036506115550016T_real(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3912_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_bool,Xsi2: list_bool,A2: fun_VE3808733824349961398l_assn,A8: fun_VE3808733824349961398l_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( aa_list_bool_nat(size_size_list_bool,Xsi) = aa_list_bool_nat(size_size_list_bool,Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = aa_list_bool_nat(size_size_list_bool,Xsi) )
                   => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                      & ( pp(aa_nat_bool(nth_bool(Xsi),I2))
                      <=> pp(aa_nat_bool(nth_bool(Xsi2),I2)) )
                      & ( aa_bool_assn(aa_VEB1972364096475303967l_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)),aa_nat_bool(nth_bool(Xsi),I2)) = aa_bool_assn(aa_VEB1972364096475303967l_assn(A8,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_bool(nth_bool(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L2302322349616425472i_bool(I5,A2,Xs,Xsi) = vEBT_L2302322349616425472i_bool(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3913_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_bool,Xsi2: list_bool,A2: fun_VE1323945509527881313l_assn,A8: fun_VE1323945509527881313l_assn] :
      ( ( I5 = I6 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( aa_list_bool_nat(size_size_list_bool,Xsi) = aa_list_bool_nat(size_size_list_bool,Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_bool_nat(size_size_list_bool,Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                      & ( pp(aa_nat_bool(nth_bool(Xsi),I2))
                      <=> pp(aa_nat_bool(nth_bool(Xsi2),I2)) )
                      & ( aa_bool_assn(aa_VEB2879522196611654106l_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)),aa_nat_bool(nth_bool(Xsi),I2)) = aa_bool_assn(aa_VEB2879522196611654106l_assn(A8,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2)),aa_nat_bool(nth_bool(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L6176004506706552451T_bool(I5,A2,Xs,Xsi) = vEBT_L6176004506706552451T_bool(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3914_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_nat,Xsi2: list_nat,A2: fun_VE1225457145571040039t_assn,A8: fun_VE1225457145571040039t_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( aa_list_nat_nat(size_size_list_nat,Xsi) = aa_list_nat_nat(size_size_list_nat,Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = aa_list_nat_nat(size_size_list_nat,Xsi) )
                   => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                      & ( aa_nat_nat(nth_nat(Xsi),I2) = aa_nat_nat(nth_nat(Xsi2),I2) )
                      & ( aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)),aa_nat_nat(nth_nat(Xsi),I2)) = aa_nat_assn(aa_VEB8068455526268438248t_assn(A8,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_nat(nth_nat(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi) = vEBT_L2809031099982602151Ti_nat(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3915_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A2: fun_VE4931415397359990460t_assn,A8: fun_VE4931415397359990460t_assn] :
      ( ( I5 = I6 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( aa_list_nat_nat(size_size_list_nat,Xsi) = aa_list_nat_nat(size_size_list_nat,Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2) )
                      & ( aa_nat_nat(nth_nat(Xsi),I2) = aa_nat_nat(nth_nat(Xsi2),I2) )
                      & ( aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)),aa_nat_nat(nth_nat(Xsi),I2)) = aa_nat_assn(aa_VEB8640158415945897581t_assn(A8,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I2)),aa_nat_nat(nth_nat(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi) = vEBT_L8650695023172932196BT_nat(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3916_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_int,Xsi2: list_int,A2: fun_VE5641251438918192003t_assn,A8: fun_VE5641251438918192003t_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( aa_list_int_nat(size_size_list_int,Xsi) = aa_list_int_nat(size_size_list_int,Xsi2) )
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = aa_list_int_nat(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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)),aa_nat_int(nth_int(Xsi),I2)) = aa_int_assn(aa_VEB9043990991474615620t_assn(A8,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_int(nth_int(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L2806540629473551875Ti_int(I5,A2,Xs,Xsi) = vEBT_L2806540629473551875Ti_int(I6,A8,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_3917_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_assn] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_assn(G,X) = one_one_assn ) )
       => ( groups3916077543535352520t_assn(G,S3) = groups3916077543535352520t_assn(G,T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3918_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_real(G,X) = one_one_real ) )
       => ( aa_set_nat_real(groups2144956460914285644t_real(G),S3) = aa_set_nat_real(groups2144956460914285644t_real(G),T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3919_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_rat(G,X) = one_one_rat ) )
       => ( groups4112464933559648120at_rat(G,S3) = groups4112464933559648120at_rat(G,T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3920_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_nat(G,X) = one_one_nat ) )
       => ( aa_set_nat_nat(groups4747594993646143856at_nat(G),S3) = aa_set_nat_nat(groups4747594993646143856at_nat(G),T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3921_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_int(G,X) = one_one_int ) )
       => ( groups4745104523137093580at_int(G,S3) = groups4745104523137093580at_int(G,T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3922_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_assn] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_assn(G,X) = one_one_assn ) )
       => ( groups4891613008741529892t_assn(G,S3) = groups4891613008741529892t_assn(G,T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3923_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_real(G,X) = one_one_real ) )
       => ( groups4331878035607307432t_real(G,S3) = groups4331878035607307432t_real(G,T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3924_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_rat(G,X) = one_one_rat ) )
       => ( groups5111818645460702676nt_rat(G,S3) = groups5111818645460702676nt_rat(G,T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3925_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_nat(G,X) = one_one_nat ) )
       => ( groups5746948705547198412nt_nat(G,S3) = groups5746948705547198412nt_nat(G,T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3926_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_int(G,X) = one_one_int ) )
       => ( aa_set_int_int(groups5744458235038148136nt_int(G),S3) = aa_set_int_int(groups5744458235038148136nt_int(G),T3) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3927_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_assn] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_assn(G,X) = one_one_assn ) )
       => ( groups3916077543535352520t_assn(G,T3) = groups3916077543535352520t_assn(G,S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3928_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_real(G,X) = one_one_real ) )
       => ( aa_set_nat_real(groups2144956460914285644t_real(G),T3) = aa_set_nat_real(groups2144956460914285644t_real(G),S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3929_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_rat(G,X) = one_one_rat ) )
       => ( groups4112464933559648120at_rat(G,T3) = groups4112464933559648120at_rat(G,S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3930_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_nat(G,X) = one_one_nat ) )
       => ( aa_set_nat_nat(groups4747594993646143856at_nat(G),T3) = aa_set_nat_nat(groups4747594993646143856at_nat(G),S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3931_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),T3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T3),S3)))
           => ( aa_nat_int(G,X) = one_one_int ) )
       => ( groups4745104523137093580at_int(G,T3) = groups4745104523137093580at_int(G,S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3932_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_assn] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_assn(G,X) = one_one_assn ) )
       => ( groups4891613008741529892t_assn(G,T3) = groups4891613008741529892t_assn(G,S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3933_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_real(G,X) = one_one_real ) )
       => ( groups4331878035607307432t_real(G,T3) = groups4331878035607307432t_real(G,S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3934_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_rat(G,X) = one_one_rat ) )
       => ( groups5111818645460702676nt_rat(G,T3) = groups5111818645460702676nt_rat(G,S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3935_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_nat(G,X) = one_one_nat ) )
       => ( groups5746948705547198412nt_nat(G,T3) = groups5746948705547198412nt_nat(G,S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3936_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T3),S3)))
           => ( aa_int_int(G,X) = one_one_int ) )
       => ( aa_set_int_int(groups5744458235038148136nt_int(G),T3) = aa_set_int_int(groups5744458235038148136nt_int(G),S3) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3937_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VEBT_VEBT_assn,G: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_assn(H,I2) = one_one_assn ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_assn(G,X) = aa_VEBT_VEBT_assn(H,X) ) )
         => ( groups8339234656476930702T_assn(G,S3) = groups8339234656476930702T_assn(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3938_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_assn,G: fun_real_assn] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_assn(H,I2) = one_one_assn ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_assn(G,X) = aa_real_assn(H,X) ) )
         => ( groups5834761644390034980l_assn(G,S3) = groups5834761644390034980l_assn(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3939_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_real(H,I2) = one_one_real ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_real(G,X) = aa_VEBT_VEBT_real(H,X) ) )
         => ( groups5615669421813869842T_real(G,S3) = groups5615669421813869842T_real(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3940_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_real(H,I2) = one_one_real ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
         => ( groups1542684614274415016l_real(G,S3) = groups1542684614274415016l_real(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3941_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_rat(H,I2) = one_one_rat ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_rat(G,X) = aa_VEBT_VEBT_rat(H,X) ) )
         => ( groups7596523497119011646BT_rat(G,S3) = groups7596523497119011646BT_rat(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3942_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_rat(H,I2) = one_one_rat ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_rat(G,X) = aa_real_rat(H,X) ) )
         => ( groups6077134973956688596al_rat(G,S3) = groups6077134973956688596al_rat(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3943_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_nat(H,I2) = one_one_nat ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_nat(G,X) = aa_VEBT_VEBT_nat(H,X) ) )
         => ( groups8231653557205507382BT_nat(G,S3) = groups8231653557205507382BT_nat(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3944_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_nat(H,I2) = one_one_nat ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_nat(G,X) = aa_real_nat(H,X) ) )
         => ( groups6712265034043184332al_nat(G,S3) = groups6712265034043184332al_nat(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3945_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_int(H,I2) = one_one_int ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_int(G,X) = aa_VEBT_VEBT_int(H,X) ) )
         => ( groups8229163086696457106BT_int(G,S3) = groups8229163086696457106BT_int(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3946_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H: fun_real_int,G: fun_real_int] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),minus_minus_set_real(T3,S3)))
           => ( aa_real_int(H,I2) = one_one_int ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_int(G,X) = aa_real_int(H,X) ) )
         => ( groups6709774563534134056al_int(G,S3) = groups6709774563534134056al_int(H,T3) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3947_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,H: fun_VEBT_VEBT_assn] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_assn(G,X) = one_one_assn ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_assn(G,X) = aa_VEBT_VEBT_assn(H,X) ) )
         => ( groups8339234656476930702T_assn(G,T3) = groups8339234656476930702T_assn(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3948_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_assn,H: fun_real_assn] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_assn(G,X) = one_one_assn ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_assn(G,X) = aa_real_assn(H,X) ) )
         => ( groups5834761644390034980l_assn(G,T3) = groups5834761644390034980l_assn(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3949_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_real(G,X) = one_one_real ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_real(G,X) = aa_VEBT_VEBT_real(H,X) ) )
         => ( groups5615669421813869842T_real(G,T3) = groups5615669421813869842T_real(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3950_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_real(G,X) = one_one_real ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
         => ( groups1542684614274415016l_real(G,T3) = groups1542684614274415016l_real(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3951_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_rat(G,X) = one_one_rat ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_rat(G,X) = aa_VEBT_VEBT_rat(H,X) ) )
         => ( groups7596523497119011646BT_rat(G,T3) = groups7596523497119011646BT_rat(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3952_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_rat(G,X) = one_one_rat ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_rat(G,X) = aa_real_rat(H,X) ) )
         => ( groups6077134973956688596al_rat(G,T3) = groups6077134973956688596al_rat(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3953_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,H: fun_VEBT_VEBT_nat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_nat(G,X) = one_one_nat ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_nat(G,X) = aa_VEBT_VEBT_nat(H,X) ) )
         => ( groups8231653557205507382BT_nat(G,T3) = groups8231653557205507382BT_nat(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3954_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_nat(G,X) = one_one_nat ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_nat(G,X) = aa_real_nat(H,X) ) )
         => ( groups6712265034043184332al_nat(G,T3) = groups6712265034043184332al_nat(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3955_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,H: fun_VEBT_VEBT_int] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(S3),T3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T3,S3)))
           => ( aa_VEBT_VEBT_int(G,X) = one_one_int ) )
       => ( ! [X: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),S3))
             => ( aa_VEBT_VEBT_int(G,X) = aa_VEBT_VEBT_int(H,X) ) )
         => ( groups8229163086696457106BT_int(G,T3) = groups8229163086696457106BT_int(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3956_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: fun_real_int,H: fun_real_int] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(S3),T3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),minus_minus_set_real(T3,S3)))
           => ( aa_real_int(G,X) = one_one_int ) )
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),S3))
             => ( aa_real_int(G,X) = aa_real_int(H,X) ) )
         => ( groups6709774563534134056al_int(G,T3) = groups6709774563534134056al_int(H,S3) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3957_sum_Odistrib_H,axiom,
    ! [I5: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_ep(I5),G))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_ep(I5),H))))
       => ( groups3521240112447731263omplex(aa_fun8982142231764154876omplex(aTP_Lamm_my(G),H),I5) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups3521240112447731263omplex(G,I5)),groups3521240112447731263omplex(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_3958_sum_Odistrib_H,axiom,
    ! [I5: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_er(I5),G))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_er(I5),H))))
       => ( groups5683813829254066239omplex(aa_fun2233226938935547116omplex(aTP_Lamm_mz(G),H),I5) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups5683813829254066239omplex(G,I5)),groups5683813829254066239omplex(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_3959_sum_Odistrib_H,axiom,
    ! [I5: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_et(I5),G))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_et(I5),H))))
       => ( groups8515261248781899619omplex(aa_fun2955221450537884980omplex(aTP_Lamm_na(G),H),I5) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups8515261248781899619omplex(G,I5)),groups8515261248781899619omplex(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_3960_sum_Odistrib_H,axiom,
    ! [I5: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_ev(I5),G))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_ev(I5),H))))
       => ( groups267424677133301183omplex(aa_fun3088484340408535532omplex(aTP_Lamm_nb(G),H),I5) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups267424677133301183omplex(G,I5)),groups267424677133301183omplex(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_3961_sum_Odistrib_H,axiom,
    ! [I5: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ex(I5),G))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_ex(I5),H))))
       => ( groups808145749697022017omplex(aa_fun3187328805393345264omplex(aTP_Lamm_nc(G),H),I5) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups808145749697022017omplex(G,I5)),groups808145749697022017omplex(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_3962_sum_Odistrib_H,axiom,
    ! [I5: set_Code_integer,G: fun_Co1900716531650368264omplex,H: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aTP_Lamm_ez(I5),G))))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun5910925830913328781r_bool(aTP_Lamm_ez(I5),H))))
       => ( groups3262226078671967728omplex(aa_fun7600406554973022798omplex(aTP_Lamm_nd(G),H),I5) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups3262226078671967728omplex(G,I5)),groups3262226078671967728omplex(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_3963_sum_Odistrib_H,axiom,
    ! [I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_fb(I5),G))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_fb(I5),H))))
       => ( groups7811786883911161277T_real(aa_fun1825138017709814396T_real(aTP_Lamm_ne(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups7811786883911161277T_real(G,I5)),groups7811786883911161277T_real(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_3964_sum_Odistrib_H,axiom,
    ! [I5: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fd(I5),G))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_fd(I5),H))))
       => ( groups97945582718554045l_real(aa_fun6946558556880649704l_real(aTP_Lamm_nf(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups97945582718554045l_real(G,I5)),groups97945582718554045l_real(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_3965_sum_Odistrib_H,axiom,
    ! [I5: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ff(I5),G))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ff(I5),H))))
       => ( aa_set_nat_real(groups8560362682196896993t_real(aa_fun1839206346889086768t_real(aTP_Lamm_me(G),H)),I5) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups8560362682196896993t_real(G),I5)),aa_set_nat_real(groups8560362682196896993t_real(H),I5)) ) ) ) ).

% sum.distrib'
tff(fact_3966_sum_Odistrib_H,axiom,
    ! [I5: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_fh(I5),G))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_fh(I5),H))))
       => ( groups1523912220035142973t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mf(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups1523912220035142973t_real(G,I5)),groups1523912220035142973t_real(H,I5)) ) ) ) ).

% sum.distrib'
tff(fact_3967_length__removeAll__less,axiom,
    ! [X3: set_nat,Xs: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s3254054031482475050et_nat(removeAll_set_nat(X3,Xs))),size_s3254054031482475050et_nat(Xs))) ) ).

% length_removeAll_less
tff(fact_3968_length__removeAll__less,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(removeAll_VEBT_VEBT(X3,Xs))),size_s6755466524823107622T_VEBT(Xs))) ) ).

% length_removeAll_less
tff(fact_3969_length__removeAll__less,axiom,
    ! [X3: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,removeAll_real(X3,Xs))),aa_list_real_nat(size_size_list_real,Xs))) ) ).

% length_removeAll_less
tff(fact_3970_length__removeAll__less,axiom,
    ! [X3: bool,Xs: list_bool] :
      ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_bool_nat(size_size_list_bool,removeAll_bool(X3,Xs))),aa_list_bool_nat(size_size_list_bool,Xs))) ) ).

% length_removeAll_less
tff(fact_3971_length__removeAll__less,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_nat_nat(size_size_list_nat,removeAll_nat(X3,Xs))),aa_list_nat_nat(size_size_list_nat,Xs))) ) ).

% length_removeAll_less
tff(fact_3972_length__removeAll__less,axiom,
    ! [X3: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_int_nat(size_size_list_int,removeAll_int(X3,Xs))),aa_list_int_nat(size_size_list_int,Xs))) ) ).

% length_removeAll_less
tff(fact_3973_prod_Odistrib_H,axiom,
    ! [I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_du(I5),G))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_du(I5),H))))
       => ( groups5615669421813869842T_real(aa_fun1825138017709814396T_real(aTP_Lamm_ng(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups5615669421813869842T_real(G,I5)),groups5615669421813869842T_real(H,I5)) ) ) ) ).

% prod.distrib'
tff(fact_3974_prod_Odistrib_H,axiom,
    ! [I5: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dw(I5),G))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dw(I5),H))))
       => ( groups1542684614274415016l_real(aa_fun6946558556880649704l_real(aTP_Lamm_nh(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups1542684614274415016l_real(G,I5)),groups1542684614274415016l_real(H,I5)) ) ) ) ).

% prod.distrib'
tff(fact_3975_prod_Odistrib_H,axiom,
    ! [I5: set_nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dy(I5),G))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_dy(I5),H))))
       => ( aa_set_nat_real(groups2144956460914285644t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mo(G),H)),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups2144956460914285644t_real(G),I5)),aa_set_nat_real(groups2144956460914285644t_real(H),I5)) ) ) ) ).

% prod.distrib'
tff(fact_3976_prod_Odistrib_H,axiom,
    ! [I5: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ea(I5),G))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ea(I5),H))))
       => ( groups4331878035607307432t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mp(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups4331878035607307432t_real(G,I5)),groups4331878035607307432t_real(H,I5)) ) ) ) ).

% prod.distrib'
tff(fact_3977_prod_Odistrib_H,axiom,
    ! [I5: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ec(I5),G))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ec(I5),H))))
       => ( groups5446087311781667882x_real(aa_fun8933909628229591020x_real(aTP_Lamm_mq(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups5446087311781667882x_real(G,I5)),groups5446087311781667882x_real(H,I5)) ) ) ) ).

% prod.distrib'
tff(fact_3978_prod_Odistrib_H,axiom,
    ! [I5: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun7028086442024493327r_bool(aTP_Lamm_ee(I5),G))))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,collect_Code_integer(aa_fun7028086442024493327r_bool(aTP_Lamm_ee(I5),H))))
       => ( groups7484263781351690841r_real(aa_fun6341089502340155082r_real(aTP_Lamm_mr(G),H),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups7484263781351690841r_real(G,I5)),groups7484263781351690841r_real(H,I5)) ) ) ) ).

% prod.distrib'
tff(fact_3979_prod_Odistrib_H,axiom,
    ! [I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1017860439744518025T_bool(aTP_Lamm_eg(I5),G))))
     => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1017860439744518025T_bool(aTP_Lamm_eg(I5),H))))
       => ( groups7596523497119011646BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_ni(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups7596523497119011646BT_rat(G,I5)),groups7596523497119011646BT_rat(H,I5)) ) ) ) ).

% prod.distrib'
tff(fact_3980_prod_Odistrib_H,axiom,
    ! [I5: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_ei(I5),G))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_ei(I5),H))))
       => ( groups6077134973956688596al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nj(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups6077134973956688596al_rat(G,I5)),groups6077134973956688596al_rat(H,I5)) ) ) ) ).

% prod.distrib'
tff(fact_3981_prod_Odistrib_H,axiom,
    ! [I5: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ek(I5),G))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_ek(I5),H))))
       => ( groups4112464933559648120at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ms(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups4112464933559648120at_rat(G,I5)),groups4112464933559648120at_rat(H,I5)) ) ) ) ).

% prod.distrib'
tff(fact_3982_prod_Odistrib_H,axiom,
    ! [I5: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_em(I5),G))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_em(I5),H))))
       => ( groups5111818645460702676nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_mt(G),H),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups5111818645460702676nt_rat(G,I5)),groups5111818645460702676nt_rat(H,I5)) ) ) ) ).

% prod.distrib'
tff(fact_3983_is__singleton__def,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(is_sin24926331636114728T_VEBT,A2))
    <=> ? [X4: vEBT_VEBT] : A2 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X4),bot_bo8194388402131092736T_VEBT) ) ).

% is_singleton_def
tff(fact_3984_is__singleton__def,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(is_singleton_real,A2))
    <=> ? [X4: real] : A2 = aa_set_real_set_real(insert_real(X4),bot_bot_set_real) ) ).

% is_singleton_def
tff(fact_3985_is__singleton__def,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(is_singleton_bool,A2))
    <=> ? [X4: bool] : A2 = aa_set_bool_set_bool(insert_bool(X4),bot_bot_set_bool) ) ).

% is_singleton_def
tff(fact_3986_is__singleton__def,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(is_singleton_nat,A2))
    <=> ? [X4: nat] : A2 = aa_set_nat_set_nat(insert_nat(X4),bot_bot_set_nat) ) ).

% is_singleton_def
tff(fact_3987_is__singleton__def,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(is_singleton_int,A2))
    <=> ? [X4: int] : A2 = aa_set_int_set_int(insert_int(X4),bot_bot_set_int) ) ).

% is_singleton_def
tff(fact_3988_is__singletonE,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(is_sin24926331636114728T_VEBT,A2))
     => ~ ! [X: vEBT_VEBT] : A2 != aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),bot_bo8194388402131092736T_VEBT) ) ).

% is_singletonE
tff(fact_3989_is__singletonE,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(is_singleton_real,A2))
     => ~ ! [X: real] : A2 != aa_set_real_set_real(insert_real(X),bot_bot_set_real) ) ).

% is_singletonE
tff(fact_3990_is__singletonE,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(is_singleton_bool,A2))
     => ~ ! [X: bool] : A2 != aa_set_bool_set_bool(insert_bool(X),bot_bot_set_bool) ) ).

% is_singletonE
tff(fact_3991_is__singletonE,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(is_singleton_nat,A2))
     => ~ ! [X: nat] : A2 != aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ).

% is_singletonE
tff(fact_3992_is__singletonE,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(is_singleton_int,A2))
     => ~ ! [X: int] : A2 != aa_set_int_set_int(insert_int(X),bot_bot_set_int) ) ).

% is_singletonE
tff(fact_3993_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE1225457145571040039t_assn,Xsi: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2809031099982602151Ti_nat(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_3994_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE2561185598974572172i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_3995_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5577048379444288083T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_3996_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5641251438918192003t_assn,Xsi: list_int] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2806540629473551875Ti_int(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_3997_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8650695023172932196BT_nat(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_3998_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_3999_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE123837653852366616t_assn,Xsi: list_int] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8648204552663881920BT_int(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB392321844297299145t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_4000_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re574865750389657758t_assn,Xsi: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L234762979517870878al_nat(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_4001_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re560840401387337987i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_4002_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re8050108611700907676T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_4003_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4004_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4005_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4006_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2806540629473551875Ti_int(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4007_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4008_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4009_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE123837653852366616t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_int,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB392321844297299145t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4010_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_re574865750389657758t_assn,Xs: list_real,I: nat,Xsi: list_nat,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L234762979517870878al_nat(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4011_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4012_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi)),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_4013_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4014_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4015_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4016_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2806540629473551875Ti_int(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4017_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4018_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,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(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4019_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE123837653852366616t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_int,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB392321844297299145t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4020_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_re574865750389657758t_assn,Xs: list_real,I: nat,Xsi: list_nat,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L234762979517870878al_nat(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4021_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,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(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4022_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F4: assn,Q: assn] :
      ( entails(P,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(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_4023_in__set__product__lists__length,axiom,
    ! [Xs: list_real,Xss: list_list_real] :
      ( member_list_real(Xs,set_list_real2(product_lists_real(Xss)))
     => ( aa_list_real_nat(size_size_list_real,Xs) = size_s6660260683639930848t_real(Xss) ) ) ).

% in_set_product_lists_length
tff(fact_4024_in__set__product__lists__length,axiom,
    ! [Xs: list_bool,Xss: list_list_bool] :
      ( member_list_bool(Xs,set_list_bool2(product_lists_bool(Xss)))
     => ( aa_list_bool_nat(size_size_list_bool,Xs) = size_s9193230429364170767t_bool(Xss) ) ) ).

% in_set_product_lists_length
tff(fact_4025_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] :
      ( pp(aa_set_list_nat_bool(member_list_nat(Xs),set_list_nat2(product_lists_nat(Xss))))
     => ( aa_list_nat_nat(size_size_list_nat,Xs) = size_s3023201423986296836st_nat(Xss) ) ) ).

% in_set_product_lists_length
tff(fact_4026_in__set__product__lists__length,axiom,
    ! [Xs: list_int,Xss: list_list_int] :
      ( member_list_int(Xs,set_list_int2(product_lists_int(Xss)))
     => ( aa_list_int_nat(size_size_list_int,Xs) = size_s533118279054570080st_int(Xss) ) ) ).

% in_set_product_lists_length
tff(fact_4027_triangle__Suc,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).

% triangle_Suc
tff(fact_4028_Id__on__set,axiom,
    ! [Xs: list_VEBT_VEBT] : id_on_VEBT_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)) = set_Pr9182192707038809660T_VEBT(map_VE1720758354293053111T_VEBT(aTP_Lamm_nk,Xs)) ).

% Id_on_set
tff(fact_4029_Id__on__set,axiom,
    ! [Xs: list_real] : id_on_real(aa_lis3769085826154824284t_real(set_real2,Xs)) = set_Pr5999470521830281550l_real(map_re3533924840734919879l_real(aTP_Lamm_nl,Xs)) ).

% Id_on_set
tff(fact_4030_Id__on__set,axiom,
    ! [Xs: list_bool] : id_on_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs)) = set_Pr2237862708767489404l_bool(map_bo2576003307432440276l_bool(aTP_Lamm_nm,Xs)) ).

% Id_on_set
tff(fact_4031_Id__on__set,axiom,
    ! [Xs: list_nat] : id_on_nat(aa_list_nat_set_nat(set_nat2,Xs)) = set_Pr5648618587558075414at_nat(map_na7298421622053143531at_nat(aTP_Lamm_nn,Xs)) ).

% Id_on_set
tff(fact_4032_Id__on__set,axiom,
    ! [Xs: list_int] : id_on_int(aa_list_int_set_int(set_int2,Xs)) = set_Pr2470121279949933262nt_int(map_in7157766398909135175nt_int(aTP_Lamm_no,Xs)) ).

% Id_on_set
tff(fact_4033_Id__on__set,axiom,
    ! [Xs: list_Code_integer] : id_on_Code_integer(set_Code_integer2(Xs)) = set_Pr920681315882439344nteger(map_Co3589949550033412536nteger(aTP_Lamm_np,Xs)) ).

% Id_on_set
tff(fact_4034_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_nq(A2),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4035_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_nr(A2),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4036_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_Code_integer,N: nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat3680331675976215733r_bool(aTP_Lamm_ns(A2),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4037_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => finite306553202115118035t_real(collect_list_real(aa_nat4570152432052498854l_bool(aTP_Lamm_nt(A2),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4038_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_bool,N: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_nu(A2),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4039_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_nv(A2),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4040_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_nw(A2),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4041_length__product__lists,axiom,
    ! [Xss: list_list_real] : size_s6660260683639930848t_real(product_lists_real(Xss)) = aa_nat_nat(foldr_nat_nat(times_times_nat,map_list_real_nat(size_size_list_real,Xss)),one_one_nat) ).

% length_product_lists
tff(fact_4042_length__product__lists,axiom,
    ! [Xss: list_list_bool] : size_s9193230429364170767t_bool(product_lists_bool(Xss)) = aa_nat_nat(foldr_nat_nat(times_times_nat,map_list_bool_nat(size_size_list_bool,Xss)),one_one_nat) ).

% length_product_lists
tff(fact_4043_length__product__lists,axiom,
    ! [Xss: list_list_nat] : size_s3023201423986296836st_nat(product_lists_nat(Xss)) = aa_nat_nat(foldr_nat_nat(times_times_nat,map_list_nat_nat(size_size_list_nat,Xss)),one_one_nat) ).

% length_product_lists
tff(fact_4044_length__product__lists,axiom,
    ! [Xss: list_list_int] : size_s533118279054570080st_int(product_lists_int(Xss)) = aa_nat_nat(foldr_nat_nat(times_times_nat,map_list_int_nat(size_size_list_int,Xss)),one_one_nat) ).

% length_product_lists
tff(fact_4045_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_4046_ent__pure__pre__iff,axiom,
    ! [P: assn,B: bool,Q: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B)),Q)
    <=> ( pp(B)
       => entails(P,Q) ) ) ).

% ent_pure_pre_iff
tff(fact_4047_Id__on__empty,axiom,
    id_on_real(bot_bot_set_real) = bot_bo3948376660626123781l_real ).

% Id_on_empty
tff(fact_4048_Id__on__empty,axiom,
    id_on_bool(bot_bot_set_bool) = bot_bo3663083580490196787l_bool ).

% Id_on_empty
tff(fact_4049_Id__on__empty,axiom,
    id_on_nat(bot_bot_set_nat) = bot_bo2099793752762293965at_nat ).

% Id_on_empty
tff(fact_4050_Id__on__empty,axiom,
    id_on_int(bot_bot_set_int) = bot_bo1796632182523588997nt_int ).

% Id_on_empty
tff(fact_4051_ent__pure__pre__iff__sng,axiom,
    ! [B: bool,Q: assn] :
      ( entails(aa_bool_assn(pure_assn,B),Q)
    <=> ( pp(B)
       => entails(one_one_assn,Q) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_4052_ent__iffI,axiom,
    ! [A2: assn,B3: assn] :
      ( entails(A2,B3)
     => ( entails(B3,A2)
       => ( A2 = B3 ) ) ) ).

% ent_iffI
tff(fact_4053_ent__refl,axiom,
    ! [P: assn] : entails(P,P) ).

% ent_refl
tff(fact_4054_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(P,Q)
     => ( entails(Q,R)
       => entails(P,R) ) ) ).

% ent_trans
tff(fact_4055_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
     => entails(P,Q) ) ).

% is_entails
tff(fact_4056_fr__rot,axiom,
    ! [A2: assn,B3: assn,C2: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B3),C2)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B3),A2),C2) ) ).

% fr_rot
tff(fact_4057_fr__refl,axiom,
    ! [A2: assn,B3: assn,C2: assn] :
      ( entails(A2,B3)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B3),C2)) ) ).

% fr_refl
tff(fact_4058_fr__rot__rhs,axiom,
    ! [A2: assn,B3: assn,C2: assn] :
      ( entails(A2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B3),C2))
     => entails(A2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),B3)) ) ).

% fr_rot_rhs
tff(fact_4059_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q2: assn] :
      ( entails(P,P2)
     => ( entails(Q,Q2)
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Q2)) ) ) ).

% ent_star_mono
tff(fact_4060_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F4: assn,Q: assn] :
      ( entails(P,R)
     => ( entails(Ps,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F4))
       => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),F4),Q)
         => entails(Ps,Q) ) ) ) ).

% ent_frame_fwd
tff(fact_4061_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_option_nat_assn,Q2: fun_option_nat_assn,C: heap_T2636463487746394924on_nat] :
      ( entails(P,P2)
     => ( ! [X: option_nat] : entails(aa_option_nat_assn(Q,X),aa_option_nat_assn(Q2,X))
       => ( hoare_7629718768684598413on_nat(P2,C,Q)
         => hoare_7629718768684598413on_nat(P,C,Q2) ) ) ) ).

% cons_rule
tff(fact_4062_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_bool_assn,Q2: fun_bool_assn,C: heap_Time_Heap_bool] :
      ( entails(P,P2)
     => ( ! [X: bool] : entails(aa_bool_assn(Q,X),aa_bool_assn(Q2,X))
       => ( hoare_2774773592644289642e_bool(P2,C,Q)
         => hoare_2774773592644289642e_bool(P,C,Q2) ) ) ) ).

% cons_rule
tff(fact_4063_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_VEBT_VEBTi_assn,Q2: fun_VEBT_VEBTi_assn,C: heap_T8145700208782473153_VEBTi] :
      ( entails(P,P2)
     => ( ! [X: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q,X),aa_VEBT_VEBTi_assn(Q2,X))
       => ( hoare_1429296392585015714_VEBTi(P2,C,Q)
         => hoare_1429296392585015714_VEBTi(P,C,Q2) ) ) ) ).

% cons_rule
tff(fact_4064_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_nat_assn,Q2: fun_nat_assn,C: heap_Time_Heap_nat] :
      ( entails(P,P2)
     => ( ! [X: nat] : entails(aa_nat_assn(Q,X),aa_nat_assn(Q2,X))
       => ( hoare_3067605981109127869le_nat(P2,C,Q)
         => hoare_3067605981109127869le_nat(P,C,Q2) ) ) ) ).

% cons_rule
tff(fact_4065_cons__post__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Q2: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( ! [X: option_nat] : entails(aa_option_nat_assn(Q,X),aa_option_nat_assn(Q2,X))
       => hoare_7629718768684598413on_nat(P,C,Q2) ) ) ).

% cons_post_rule
tff(fact_4066_cons__post__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,Q2: fun_bool_assn] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => ( ! [X: bool] : entails(aa_bool_assn(Q,X),aa_bool_assn(Q2,X))
       => hoare_2774773592644289642e_bool(P,C,Q2) ) ) ).

% cons_post_rule
tff(fact_4067_cons__post__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Q2: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( ! [X: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q,X),aa_VEBT_VEBTi_assn(Q2,X))
       => hoare_1429296392585015714_VEBTi(P,C,Q2) ) ) ).

% cons_post_rule
tff(fact_4068_cons__post__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,Q2: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( ! [X: nat] : entails(aa_nat_assn(Q,X),aa_nat_assn(Q2,X))
       => hoare_3067605981109127869le_nat(P,C,Q2) ) ) ).

% cons_post_rule
tff(fact_4069_ent__false,axiom,
    ! [P: assn] : entails(bot_bot_assn,P) ).

% ent_false
tff(fact_4070_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( entails(P,P2)
     => ( hoare_7629718768684598413on_nat(P2,C,Q)
       => hoare_7629718768684598413on_nat(P,C,Q) ) ) ).

% cons_pre_rule
tff(fact_4071_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,P2)
     => ( hoare_2774773592644289642e_bool(P2,C,Q)
       => hoare_2774773592644289642e_bool(P,C,Q) ) ) ).

% cons_pre_rule
tff(fact_4072_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( entails(P,P2)
     => ( hoare_1429296392585015714_VEBTi(P2,C,Q)
       => hoare_1429296392585015714_VEBTi(P,C,Q) ) ) ).

% cons_pre_rule
tff(fact_4073_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( entails(P,P2)
     => ( hoare_3067605981109127869le_nat(P2,C,Q)
       => hoare_3067605981109127869le_nat(P,C,Q) ) ) ).

% cons_pre_rule
tff(fact_4074_fi__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ps: assn,F4: assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( entails(Ps,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F4))
       => hoare_7629718768684598413on_nat(Ps,C,aa_ass4112278883933122603t_assn(aTP_Lamm_aa(Q),F4)) ) ) ).

% fi_rule
tff(fact_4075_fi__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,Ps: assn,F4: assn] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => ( entails(Ps,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F4))
       => hoare_2774773592644289642e_bool(Ps,C,aa_ass2376637134808068204l_assn(aTP_Lamm_ab(Q),F4)) ) ) ).

% fi_rule
tff(fact_4076_fi__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ps: assn,F4: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( entails(Ps,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F4))
       => hoare_1429296392585015714_VEBTi(Ps,C,aa_ass4389913640333755968i_assn(aTP_Lamm_ac(Q),F4)) ) ) ).

% fi_rule
tff(fact_4077_fi__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,Ps: assn,F4: assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( entails(Ps,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F4))
       => hoare_3067605981109127869le_nat(Ps,C,aa_assn_fun_nat_assn(aTP_Lamm_ad(Q),F4)) ) ) ).

% fi_rule
tff(fact_4078_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_T2636463487746394924on_nat,Q2: fun_option_nat_assn,T4: nat,P: assn,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(P2,C,Q2),T4))
     => ( entails(P,P2)
       => ( ! [X: option_nat] : entails(aa_option_nat_assn(Q2,X),aa_option_nat_assn(Q,X))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T4),T))
           => pp(aa_nat_bool(time_htt_option_nat(P,C,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_4079_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,T4: nat,P: assn,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(P2,C,Q2),T4))
     => ( entails(P,P2)
       => ( ! [X: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q2,X),aa_VEBT_VEBTi_assn(Q,X))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T4),T))
           => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,C,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_4080_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_Time_Heap_bool,Q2: fun_bool_assn,T4: nat,P: assn,Q: fun_bool_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_bool(P2,C,Q2),T4))
     => ( entails(P,P2)
       => ( ! [X: bool] : entails(aa_bool_assn(Q2,X),aa_bool_assn(Q,X))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T4),T))
           => pp(aa_nat_bool(time_htt_bool(P,C,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_4081_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_Time_Heap_nat,Q2: fun_nat_assn,T4: nat,P: assn,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(P2,C,Q2),T4))
     => ( entails(P,P2)
       => ( ! [X: nat] : entails(aa_nat_assn(Q2,X),aa_nat_assn(Q,X))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T4),T))
           => pp(aa_nat_bool(time_htt_nat(P,C,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_4082_distinct__finite__set,axiom,
    ! [X3: set_VEBT_VEBT] : finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aTP_Lamm_nx(X3))) ).

% distinct_finite_set
tff(fact_4083_distinct__finite__set,axiom,
    ! [X3: set_nat] : finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_ny(X3))) ).

% distinct_finite_set
tff(fact_4084_distinct__finite__set,axiom,
    ! [X3: set_real] : finite306553202115118035t_real(collect_list_real(aTP_Lamm_nz(X3))) ).

% distinct_finite_set
tff(fact_4085_distinct__finite__set,axiom,
    ! [X3: set_bool] : finite7866827932528546246t_bool(collect_list_bool(aTP_Lamm_oa(X3))) ).

% distinct_finite_set
tff(fact_4086_strict__sorted__iff,axiom,
    ! [L: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_real),L))
    <=> ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),L))
        & pp(aa_list_real_bool(distinct_real,L)) ) ) ).

% strict_sorted_iff
tff(fact_4087_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_4088_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_4089_strict__sorted__iff,axiom,
    ! [L: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),L))
    <=> ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
        & pp(aa_list_nat_bool(distinct_nat,L)) ) ) ).

% strict_sorted_iff
tff(fact_4090_strict__sorted__iff,axiom,
    ! [L: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),L))
    <=> ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
        & pp(aa_list_int_bool(distinct_int,L)) ) ) ).

% strict_sorted_iff
tff(fact_4091_distinct__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBTi] :
      ( distinct_VEBT_VEBTi(Xs)
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
         => ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4092_distinct__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( distinct_VEBT_VEBT(Xs)
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
         => ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4093_distinct__conv__nth,axiom,
    ! [Xs: list_real] :
      ( pp(aa_list_real_bool(distinct_real,Xs))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_real_nat(size_size_list_real,Xs)))
         => ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_real_nat(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_4094_distinct__conv__nth,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_bool_nat(size_size_list_bool,Xs)))
         => ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_bool_nat(size_size_list_bool,Xs)))
             => ( ( I3 != J3 )
               => ~ ( pp(aa_nat_bool(nth_bool(Xs),I3))
                  <=> pp(aa_nat_bool(nth_bool(Xs),J3)) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_4095_distinct__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs)))
         => ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4096_distinct__conv__nth,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,Xs)))
         => ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),aa_list_int_nat(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_4097_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,J: nat] :
      ( distinct_VEBT_VEBTi(Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4098_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,J: nat] :
      ( distinct_VEBT_VEBT(Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
         => ( ( 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_4099_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_real,I: nat,J: nat] :
      ( pp(aa_list_real_bool(distinct_real,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_real_nat(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_4100_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_bool,I: nat,J: nat] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,Xs)))
         => ( ( pp(aa_nat_bool(nth_bool(Xs),I))
            <=> pp(aa_nat_bool(nth_bool(Xs),J)) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_4101_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4102_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(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_4103_distinct__length__le,axiom,
    ! [Ys: list_VEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( distinct_VEBT_VEBT(Ys)
     => ( ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Ys)),size_s6755466524823107622T_VEBT(Xs))) ) ) ).

% distinct_length_le
tff(fact_4104_distinct__length__le,axiom,
    ! [Ys: list_real,Xs: list_real] :
      ( pp(aa_list_real_bool(distinct_real,Ys))
     => ( ( aa_lis3769085826154824284t_real(set_real2,Ys) = aa_lis3769085826154824284t_real(set_real2,Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Ys)),aa_list_real_nat(size_size_list_real,Xs))) ) ) ).

% distinct_length_le
tff(fact_4105_distinct__length__le,axiom,
    ! [Ys: list_bool,Xs: list_bool] :
      ( pp(aa_list_bool_bool(distinct_bool,Ys))
     => ( ( aa_lis3587183946281582166t_bool(set_bool2,Ys) = aa_lis3587183946281582166t_bool(set_bool2,Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Ys)),aa_list_bool_nat(size_size_list_bool,Xs))) ) ) ).

% distinct_length_le
tff(fact_4106_distinct__length__le,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Ys))
     => ( ( aa_list_nat_set_nat(set_nat2,Ys) = aa_list_nat_set_nat(set_nat2,Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4107_distinct__length__le,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,Ys))
     => ( ( aa_list_int_set_int(set_int2,Ys) = aa_list_int_set_int(set_int2,Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Ys)),aa_list_int_nat(size_size_list_int,Xs))) ) ) ).

% distinct_length_le
tff(fact_4108_sorted__distinct__set__unique,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs))
     => ( pp(aa_list_real_bool(distinct_real,Xs))
       => ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Ys))
         => ( pp(aa_list_real_bool(distinct_real,Ys))
           => ( ( aa_lis3769085826154824284t_real(set_real2,Xs) = aa_lis3769085826154824284t_real(set_real2,Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_4109_sorted__distinct__set__unique,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
     => ( pp(aa_list_bool_bool(distinct_bool,Xs))
       => ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Ys))
         => ( pp(aa_list_bool_bool(distinct_bool,Ys))
           => ( ( aa_lis3587183946281582166t_bool(set_bool2,Xs) = aa_lis3587183946281582166t_bool(set_bool2,Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_4110_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_4111_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_4112_sorted__distinct__set__unique,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => ( pp(aa_list_nat_bool(distinct_nat,Xs))
       => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Ys))
         => ( pp(aa_list_nat_bool(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_4113_sorted__distinct__set__unique,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => ( pp(aa_list_int_bool(distinct_int,Xs))
       => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Ys))
         => ( pp(aa_list_int_bool(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_4114_finite__sorted__distinct__unique,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ? [X: list_real] :
          ( ( aa_lis3769085826154824284t_real(set_real2,X) = A2 )
          & pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),X))
          & pp(aa_list_real_bool(distinct_real,X))
          & ! [Y4: list_real] :
              ( ( ( aa_lis3769085826154824284t_real(set_real2,Y4) = A2 )
                & pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Y4))
                & pp(aa_list_real_bool(distinct_real,Y4)) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_4115_finite__sorted__distinct__unique,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ? [X: list_bool] :
          ( ( aa_lis3587183946281582166t_bool(set_bool2,X) = A2 )
          & pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),X))
          & pp(aa_list_bool_bool(distinct_bool,X))
          & ! [Y4: list_bool] :
              ( ( ( aa_lis3587183946281582166t_bool(set_bool2,Y4) = A2 )
                & pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Y4))
                & pp(aa_list_bool_bool(distinct_bool,Y4)) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_4116_finite__sorted__distinct__unique,axiom,
    ! [A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ? [X: list_Code_integer] :
          ( ( set_Code_integer2(X) = A2 )
          & sorted710888440204495920nteger(ord_le3102999989581377725nteger,X)
          & distin1543349897113766820nteger(X)
          & ! [Y4: list_Code_integer] :
              ( ( ( set_Code_integer2(Y4) = A2 )
                & sorted710888440204495920nteger(ord_le3102999989581377725nteger,Y4)
                & distin1543349897113766820nteger(Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_4117_finite__sorted__distinct__unique,axiom,
    ! [A2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ? [X: list_rat] :
          ( ( set_rat2(X) = A2 )
          & sorted_wrt_rat(ord_less_eq_rat,X)
          & distinct_rat(X)
          & ! [Y4: list_rat] :
              ( ( ( set_rat2(Y4) = A2 )
                & sorted_wrt_rat(ord_less_eq_rat,Y4)
                & distinct_rat(Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_4118_finite__sorted__distinct__unique,axiom,
    ! [A2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ? [X: list_num] :
          ( ( set_num2(X) = A2 )
          & sorted_wrt_num(ord_less_eq_num,X)
          & distinct_num(X)
          & ! [Y4: list_num] :
              ( ( ( set_num2(Y4) = A2 )
                & sorted_wrt_num(ord_less_eq_num,Y4)
                & distinct_num(Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_4119_finite__sorted__distinct__unique,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ? [X: list_nat] :
          ( ( aa_list_nat_set_nat(set_nat2,X) = A2 )
          & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),X))
          & pp(aa_list_nat_bool(distinct_nat,X))
          & ! [Y4: list_nat] :
              ( ( ( aa_list_nat_set_nat(set_nat2,Y4) = A2 )
                & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Y4))
                & pp(aa_list_nat_bool(distinct_nat,Y4)) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_4120_finite__sorted__distinct__unique,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ? [X: list_int] :
          ( ( aa_list_int_set_int(set_int2,X) = A2 )
          & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),X))
          & pp(aa_list_int_bool(distinct_int,X))
          & ! [Y4: list_int] :
              ( ( ( aa_list_int_set_int(set_int2,Y4) = A2 )
                & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Y4))
                & pp(aa_list_int_bool(distinct_int,Y4)) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_4121_distinct__Ex1,axiom,
    ! [Xs: list_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( distinct_VEBT_VEBTi(Xs)
     => ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))
       => ? [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),size_s7982070591426661849_VEBTi(Xs)))
            & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),X) = X3 )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),size_s7982070591426661849_VEBTi(Xs)))
                  & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Y4) = X3 ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4122_distinct__Ex1,axiom,
    ! [Xs: list_set_nat,X3: set_nat] :
      ( distinct_set_nat(Xs)
     => ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
       => ? [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),size_s3254054031482475050et_nat(Xs)))
            & ( aa_nat_set_nat(nth_set_nat(Xs),X) = X3 )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),size_s3254054031482475050et_nat(Xs)))
                  & ( aa_nat_set_nat(nth_set_nat(Xs),Y4) = X3 ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4123_distinct__Ex1,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( distinct_VEBT_VEBT(Xs)
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
       => ? [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),size_s6755466524823107622T_VEBT(Xs)))
            & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),X) = X3 )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),size_s6755466524823107622T_VEBT(Xs)))
                  & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Y4) = X3 ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4124_distinct__Ex1,axiom,
    ! [Xs: list_real,X3: real] :
      ( pp(aa_list_real_bool(distinct_real,Xs))
     => ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
       => ? [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_list_real_nat(size_size_list_real,Xs)))
            & ( aa_nat_real(nth_real(Xs),X) = X3 )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),aa_list_real_nat(size_size_list_real,Xs)))
                  & ( aa_nat_real(nth_real(Xs),Y4) = X3 ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4125_distinct__Ex1,axiom,
    ! [Xs: list_bool,X3: bool] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
     => ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
       => ? [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_list_bool_nat(size_size_list_bool,Xs)))
            & ( pp(aa_nat_bool(nth_bool(Xs),X))
            <=> pp(X3) )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),aa_list_bool_nat(size_size_list_bool,Xs)))
                  & ( pp(aa_nat_bool(nth_bool(Xs),Y4))
                  <=> pp(X3) ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4126_distinct__Ex1,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
       => ? [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_list_nat_nat(size_size_list_nat,Xs)))
            & ( aa_nat_nat(nth_nat(Xs),X) = X3 )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),aa_list_nat_nat(size_size_list_nat,Xs)))
                  & ( aa_nat_nat(nth_nat(Xs),Y4) = X3 ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4127_distinct__Ex1,axiom,
    ! [Xs: list_int,X3: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
       => ? [X: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_list_int_nat(size_size_list_int,Xs)))
            & ( aa_nat_int(nth_int(Xs),X) = X3 )
            & ! [Y4: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),aa_list_int_nat(size_size_list_int,Xs)))
                  & ( aa_nat_int(nth_int(Xs),Y4) = X3 ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4128_distinct__idx,axiom,
    ! [F: fun_VEBT_VEBT_nat,L: list_VEBT_VEBT,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,map_VEBT_VEBT_nat(F,L)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(L)))
         => ( ( aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),J)) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_4129_distinct__idx,axiom,
    ! [F: fun_VEBT_VEBT_real,L: list_VEBT_VEBT,I: nat,J: nat] :
      ( pp(aa_list_real_bool(distinct_real,map_VEBT_VEBT_real(F,L)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(L)))
         => ( ( aa_VEBT_VEBT_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),J)) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_4130_distinct__idx,axiom,
    ! [F: fun_Pr4298194452408323251l_bool,L: list_P2722043638750504429l_bool,I: nat,J: nat] :
      ( pp(aa_list_bool_bool(distinct_bool,map_Pr8574941368599290932l_bool(F,L)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6988337448923680857l_bool(L)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6988337448923680857l_bool(L)))
         => ( ( pp(aa_Pro9114106320147817132l_bool(F,nth_Pr2856333732135533678l_bool(L,I)))
            <=> pp(aa_Pro9114106320147817132l_bool(F,nth_Pr2856333732135533678l_bool(L,J))) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_4131_distinct__idx,axiom,
    ! [F: fun_nat_nat,L: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,map_nat_nat(F,L)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L)))
         => ( ( aa_nat_nat(F,aa_nat_nat(nth_nat(L),I)) = aa_nat_nat(F,aa_nat_nat(nth_nat(L),J)) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_4132_distinct__idx,axiom,
    ! [F: fun_nat_bool,L: list_nat,I: nat,J: nat] :
      ( pp(aa_list_bool_bool(distinct_bool,map_nat_bool(F,L)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L)))
         => ( ( pp(aa_nat_bool(F,aa_nat_nat(nth_nat(L),I)))
            <=> pp(aa_nat_bool(F,aa_nat_nat(nth_nat(L),J))) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_4133_distinct__finite__subset,axiom,
    ! [X3: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,X3))
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aTP_Lamm_ob(X3))) ) ).

% distinct_finite_subset
tff(fact_4134_distinct__finite__subset,axiom,
    ! [X3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,X3))
     => finite306553202115118035t_real(collect_list_real(aTP_Lamm_oc(X3))) ) ).

% distinct_finite_subset
tff(fact_4135_distinct__finite__subset,axiom,
    ! [X3: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,X3))
     => finite7866827932528546246t_bool(collect_list_bool(aTP_Lamm_od(X3))) ) ).

% distinct_finite_subset
tff(fact_4136_distinct__finite__subset,axiom,
    ! [X3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,X3))
     => finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_oe(X3))) ) ).

% distinct_finite_subset
tff(fact_4137_distinct__finite__subset,axiom,
    ! [X3: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,X3))
     => finite8712137658972009173omplex(collect_list_complex(aTP_Lamm_of(X3))) ) ).

% distinct_finite_subset
tff(fact_4138_distinct__finite__subset,axiom,
    ! [X3: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,X3))
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aTP_Lamm_og(X3))) ) ).

% distinct_finite_subset
tff(fact_4139_distinct__finite__subset,axiom,
    ! [X3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,X3))
     => finite3922522038869484883st_int(collect_list_int(aTP_Lamm_oh(X3))) ) ).

% distinct_finite_subset
tff(fact_4140_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_bool,I: nat,J: nat] :
      ( pp(aa_list_bool_bool(distinct_bool,L))
     => ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,L)))
           => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,aa_nat_bool(nth_bool(L),I)),aa_nat_bool(nth_bool(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_4141_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_real,I: nat,J: nat] :
      ( pp(aa_list_real_bool(distinct_real,L))
     => ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_real_nat(size_size_list_real,L)))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(nth_real(L),I)),aa_nat_real(nth_real(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_4142_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_rat,I: nat,J: nat] :
      ( distinct_rat(L)
     => ( sorted_wrt_rat(ord_less_eq_rat,L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_rat(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_rat(L)))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(nth_rat(L),I)),aa_nat_rat(nth_rat(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_4143_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( distinct_num(L)
     => ( sorted_wrt_num(ord_less_eq_num,L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_num(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_num(L)))
           => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(nth_num(L),I)),aa_nat_num(nth_num(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_4144_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,L))
     => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L)))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(nth_nat(L),I)),aa_nat_nat(nth_nat(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_4145_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(distinct_int,L))
     => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(size_size_list_int,L)))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(nth_int(L),I)),aa_nat_int(nth_int(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_4146_distinct__sorted__mono,axiom,
    ! [L: list_bool,I: nat,J: nat] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),L))
     => ( pp(aa_list_bool_bool(distinct_bool,L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,L)))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,aa_nat_bool(nth_bool(L),I)),aa_nat_bool(nth_bool(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_4147_distinct__sorted__mono,axiom,
    ! [L: list_real,I: nat,J: nat] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),L))
     => ( pp(aa_list_real_bool(distinct_real,L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_real_nat(size_size_list_real,L)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(nth_real(L),I)),aa_nat_real(nth_real(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_4148_distinct__sorted__mono,axiom,
    ! [L: list_rat,I: nat,J: nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,L)
     => ( distinct_rat(L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_rat(L)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(nth_rat(L),I)),aa_nat_rat(nth_rat(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_4149_distinct__sorted__mono,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( sorted_wrt_num(ord_less_eq_num,L)
     => ( distinct_num(L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_num(L)))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(nth_num(L),I)),aa_nat_num(nth_num(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_4150_distinct__sorted__mono,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
     => ( pp(aa_list_nat_bool(distinct_nat,L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(nth_nat(L),I)),aa_nat_nat(nth_nat(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_4151_distinct__sorted__mono,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
     => ( pp(aa_list_int_bool(distinct_int,L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(size_size_list_int,L)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(nth_int(L),I)),aa_nat_int(nth_int(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_4152_distinct__sorted__mono__iff,axiom,
    ! [L: list_real,I: nat,J: nat] :
      ( pp(aa_list_real_bool(distinct_real,L))
     => ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_real_nat(size_size_list_real,L)))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(nth_real(L),I)),aa_nat_real(nth_real(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_4153_distinct__sorted__mono__iff,axiom,
    ! [L: list_bool,I: nat,J: nat] :
      ( pp(aa_list_bool_bool(distinct_bool,L))
     => ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,L)))
           => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(nth_bool(L),I)),aa_nat_bool(nth_bool(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_4154_distinct__sorted__mono__iff,axiom,
    ! [L: list_rat,I: nat,J: nat] :
      ( distinct_rat(L)
     => ( sorted_wrt_rat(ord_less_eq_rat,L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_rat(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_rat(L)))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(nth_rat(L),I)),aa_nat_rat(nth_rat(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_4155_distinct__sorted__mono__iff,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( distinct_num(L)
     => ( sorted_wrt_num(ord_less_eq_num,L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_num(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_num(L)))
           => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(nth_num(L),I)),aa_nat_num(nth_num(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_4156_distinct__sorted__mono__iff,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,L))
     => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L)))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(L),I)),aa_nat_nat(nth_nat(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_4157_distinct__sorted__mono__iff,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(distinct_int,L))
     => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(size_size_list_int,L)))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(L),I)),aa_nat_int(nth_int(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_4158_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,aa_lis2508312872377092907i_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))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))) ).

% assnle
tff(fact_4159_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,Xsi: list_nat,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8930081998596925642Ti_nat(A2,Xs,Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis3503319847846646691l_assn(aTP_Lamm_oi(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4160_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1891944875198410415_VEBTi(A2,Xs,Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( hoare_2774773592644289642e_bool(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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9992151435860550l_assn(aa_lis4074100605738197896l_assn(aTP_Lamm_oj(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4161_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7265847600308530106T_VEBT(A2,Xs,Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( hoare_2774773592644289642e_bool(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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis183111350483865165l_assn(aTP_Lamm_ok(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4162_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,Xsi: list_int,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8927591528087875366Ti_int(A2,Xs,Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,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(A2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis7389664606029993405l_assn(aa_lis1781686709860263423l_assn(aTP_Lamm_ol(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4163_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,Xsi: list_nat,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A2,Xs),Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,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(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis5809874432478338334l_assn(aTP_Lamm_om(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4164_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1279224858307276611T_VEBT(A2,Xs,Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_2774773592644289642e_bool(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(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,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(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis7769146593908314002l_assn(aTP_Lamm_on(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4165_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE123837653852366616t_assn,Xs: list_VEBT_VEBT,Xsi: list_int,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8294436054247626077BT_int(A2,Xs,Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB392321844297299145t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB392321844297299145t_assn(A2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis7389664606029993405l_assn(aa_lis4088241294491955066l_assn(aTP_Lamm_oo(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4166_rule__at__index,axiom,
    ! [P: assn,A2: fun_re574865750389657758t_assn,Xs: list_real,Xsi: list_nat,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1446010312343316929al_nat(A2,Xs,Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis4366942040065530138l_assn(aTP_Lamm_op(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4167_rule__at__index,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,Xs: list_real,Xsi: list_VEBT_VEBTi,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L9060850011106065574_VEBTi(A2,Xs,Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( hoare_2774773592644289642e_bool(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(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,R3),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(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9992151435860550l_assn(aa_lis3027762689479947775l_assn(aTP_Lamm_oq(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4168_rule__at__index,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,Xs: list_real,Xsi: list_VEBT_VEBT,F4: assn,I: nat,C: heap_Time_Heap_bool,Q2: fun_bool_assn,F6: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L4595930785310033027T_VEBT(A2,Xs,Xsi)),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( hoare_2774773592644289642e_bool(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(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4),C,Q2)
         => ( ! [R3: bool] : entails(aa_bool_assn(Q2,R3),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(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs))),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F6,R3)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis7705788952316199574l_assn(aTP_Lamm_or(A2),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_4169_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% recomp
tff(fact_4170_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] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% repack
tff(fact_4171_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% txe
tff(fact_4172_mergesort__remdups__correct,axiom,
    ! [L: list_real] :
      ( pp(aa_list_real_bool(distinct_real,merges7559785487730971421s_real(L)))
      & pp(aa_list_real_bool(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_4173_mergesort__remdups__correct,axiom,
    ! [L: list_bool] :
      ( pp(aa_list_bool_bool(distinct_bool,merges8832032441967277926s_bool(L)))
      & pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),merges8832032441967277926s_bool(L)))
      & ( aa_lis3587183946281582166t_bool(set_bool2,merges8832032441967277926s_bool(L)) = aa_lis3587183946281582166t_bool(set_bool2,L) ) ) ).

% mergesort_remdups_correct
tff(fact_4174_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_4175_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_4176_mergesort__remdups__correct,axiom,
    ! [L: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,merges1656613366846331073ps_nat(L)))
      & pp(aa_list_nat_bool(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_4177_mergesort__remdups__correct,axiom,
    ! [L: list_int] :
      ( pp(aa_list_int_bool(distinct_int,merges1654122896337280797ps_int(L)))
      & pp(aa_list_int_bool(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_4178_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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_4179_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,X3: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L886525131989349516_VEBTi(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4180_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,X3: vEBT_VEBTi,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2497118539674116125T_VEBT(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4181_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,X3: vEBT_VEBT,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3204528365124325536T_VEBT(I5,A2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4182_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,X3: real,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7851252805511451907_VEBTi(I5,A2,aa_real_list_real(list_update_real(Xs,I),X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4183_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,X3: real,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3095048238742455910T_VEBT(I5,A2,aa_real_list_real(list_update_real(Xs,I),X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4184_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_bo7535889243153246212i_assn,X3: bool,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBTi,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(A2,X3),Xi)),vEBT_L6372884366951783724_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6372884366951783724_VEBTi(I5,A2,list_update_bool(Xs,I,X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4185_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_bo5394993412183903963T_assn,X3: bool,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBT,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A2,X3),Xi)),vEBT_L2992182275864018813T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2992182275864018813T_VEBT(I5,A2,list_update_bool(Xs,I,X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4186_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_na6467670216208870567i_assn,X3: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L7489483478785760935_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7489483478785760935_VEBTi(I5,A2,list_update_nat(Xs,I,X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4187_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_na4489259242115493880T_assn,X3: nat,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L8511957252848910786T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8511957252848910786T_VEBT(I5,A2,list_update_nat(Xs,I,X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4188_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_in1903134756389476099i_assn,X3: int,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_int,Xsi: list_VEBT_VEBTi,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L114188773329725699_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L114188773329725699_VEBTi(I5,A2,list_update_int(Xs,I,X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_4189_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X3: vEBT_VEBTi] : size_s7982070591426661849_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X3)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_list_update
tff(fact_4190_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] : size_s6755466524823107622T_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_4191_length__list__update,axiom,
    ! [Xs: list_real,I: nat,X3: real] : aa_list_real_nat(size_size_list_real,aa_real_list_real(list_update_real(Xs,I),X3)) = aa_list_real_nat(size_size_list_real,Xs) ).

% length_list_update
tff(fact_4192_length__list__update,axiom,
    ! [Xs: list_bool,I: nat,X3: bool] : aa_list_bool_nat(size_size_list_bool,list_update_bool(Xs,I,X3)) = aa_list_bool_nat(size_size_list_bool,Xs) ).

% length_list_update
tff(fact_4193_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] : aa_list_nat_nat(size_size_list_nat,list_update_nat(Xs,I,X3)) = aa_list_nat_nat(size_size_list_nat,Xs) ).

% length_list_update
tff(fact_4194_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X3: int] : aa_list_int_nat(size_size_list_int,list_update_int(Xs,I,X3)) = aa_list_int_nat(size_size_list_int,Xs) ).

% length_list_update
tff(fact_4195_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),set_or3540276404033026485et_nat(L,U)))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,L),I))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_4196_atLeastLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I),aa_real_set_real(set_or66887138388493659n_real(L),U)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_4197_atLeastLessThan__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or8585797421378605585et_int(L,U))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(L),I))
        & pp(aa_set_int_bool(ord_less_set_int(I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_4198_atLeastLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_or4029947393144176647an_rat(L,U)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_4199_atLeastLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( pp(aa_set_num_bool(member_num(I),set_or1222409239386451017an_num(L,U)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_4200_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_4201_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),aa_int_set_int(set_or4662586982721622107an_int(L),U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_4202_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),set_or8404916559141939852nteger(L,U)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,L),I))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_4203_atLeastLessThan__empty,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( aa_real_set_real(set_or66887138388493659n_real(A),B) = bot_bot_set_real ) ) ).

% atLeastLessThan_empty
tff(fact_4204_atLeastLessThan__empty,axiom,
    ! [B: bool,A: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,B),A))
     => ( aa_bool_set_bool(set_or8576162586734908840n_bool(A),B) = bot_bot_set_bool ) ) ).

% atLeastLessThan_empty
tff(fact_4205_atLeastLessThan__empty,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( set_or8585797421378605585et_int(A,B) = bot_bot_set_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_4206_atLeastLessThan__empty,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
tff(fact_4207_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
tff(fact_4208_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_4209_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_set_int(set_or4662586982721622107an_int(A),B) = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_4210_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
     => ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
tff(fact_4211_atLeastLessThan__empty__iff,axiom,
    ! [A: bool,B: bool] :
      ( ( aa_bool_set_bool(set_or8576162586734908840n_bool(A),B) = bot_bot_set_bool )
    <=> ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_4212_atLeastLessThan__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_set_real(set_or66887138388493659n_real(A),B) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_4213_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_4214_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_4215_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_4216_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_set_int(set_or4662586982721622107an_int(A),B) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_4217_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger )
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_4218_atLeastLessThan__empty__iff2,axiom,
    ! [A: bool,B: bool] :
      ( ( bot_bot_set_bool = aa_bool_set_bool(set_or8576162586734908840n_bool(A),B) )
    <=> ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4219_atLeastLessThan__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = aa_real_set_real(set_or66887138388493659n_real(A),B) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4220_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or4029947393144176647an_rat(A,B) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4221_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or1222409239386451017an_num(A,B) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4222_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4223_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = aa_int_set_int(set_or4662586982721622107an_int(A),B) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4224_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or8404916559141939852nteger(A,B) )
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4225_infinite__Ico__iff,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,aa_real_set_real(set_or66887138388493659n_real(A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% infinite_Ico_iff
tff(fact_4226_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or4029947393144176647an_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% infinite_Ico_iff
tff(fact_4227_ivl__subset,axiom,
    ! [I: rat,J: rat,M: rat,N: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or4029947393144176647an_rat(I,J)),set_or4029947393144176647an_rat(M,N)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,J),I))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,M),I))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,J),N)) ) ) ) ).

% ivl_subset
tff(fact_4228_ivl__subset,axiom,
    ! [I: num,J: num,M: num,N: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or1222409239386451017an_num(I,J)),set_or1222409239386451017an_num(M,N)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,J),I))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),I))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,J),N)) ) ) ) ).

% ivl_subset
tff(fact_4229_ivl__subset,axiom,
    ! [I: nat,J: nat,M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I),J)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),N)) ) ) ) ).

% ivl_subset
tff(fact_4230_ivl__subset,axiom,
    ! [I: int,J: int,M: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(set_or4662586982721622107an_int(I),J)),aa_int_set_int(set_or4662586982721622107an_int(M),N)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),I))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),N)) ) ) ) ).

% ivl_subset
tff(fact_4231_ivl__subset,axiom,
    ! [I: code_integer,J: code_integer,M: code_integer,N: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(I,J)),set_or8404916559141939852nteger(M,N)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,J),I))
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,M),I))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,J),N)) ) ) ) ).

% ivl_subset
tff(fact_4232_ivl__diff,axiom,
    ! [I: rat,N: rat,M: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),N))
     => ( minus_minus_set_rat(set_or4029947393144176647an_rat(I,M),set_or4029947393144176647an_rat(I,N)) = set_or4029947393144176647an_rat(N,M) ) ) ).

% ivl_diff
tff(fact_4233_ivl__diff,axiom,
    ! [I: num,N: num,M: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),N))
     => ( minus_minus_set_num(set_or1222409239386451017an_num(I,M),set_or1222409239386451017an_num(I,N)) = set_or1222409239386451017an_num(N,M) ) ) ).

% ivl_diff
tff(fact_4234_ivl__diff,axiom,
    ! [I: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I),N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,N),M) ) ) ).

% ivl_diff
tff(fact_4235_ivl__diff,axiom,
    ! [I: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),N))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_int_set_int(set_or4662586982721622107an_int(I),M)),aa_int_set_int(set_or4662586982721622107an_int(I),N)) = aa_int_set_int(set_or4662586982721622107an_int(N),M) ) ) ).

% ivl_diff
tff(fact_4236_ivl__diff,axiom,
    ! [I: code_integer,N: code_integer,M: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),N))
     => ( minus_2355218937544613996nteger(set_or8404916559141939852nteger(I,M),set_or8404916559141939852nteger(I,N)) = set_or8404916559141939852nteger(N,M) ) ) ).

% ivl_diff
tff(fact_4237_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Xs)),I))
     => ( list_u6098035379799741383_VEBTi(Xs,I,X3) = Xs ) ) ).

% list_update_beyond
tff(fact_4238_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I))
     => ( aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3) = Xs ) ) ).

% list_update_beyond
tff(fact_4239_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),I))
     => ( aa_real_list_real(list_update_real(Xs,I),X3) = Xs ) ) ).

% list_update_beyond
tff(fact_4240_list__update__beyond,axiom,
    ! [Xs: list_bool,I: nat,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),I))
     => ( list_update_bool(Xs,I,X3) = Xs ) ) ).

% list_update_beyond
tff(fact_4241_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),I))
     => ( list_update_nat(Xs,I,X3) = Xs ) ) ).

% list_update_beyond
tff(fact_4242_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),I))
     => ( list_update_int(Xs,I,X3) = Xs ) ) ).

% list_update_beyond
tff(fact_4243_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F4: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( entails(P,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(Uu),Uua)),Xi)),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(I5,vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),C,Q)
           => hoare_7629718768684598413on_nat(P,C,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_4244_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F4: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,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(Uu),Uua)),Xi)),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(I5,vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_4245_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F4: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( entails(P,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(Uu),Uua)),Xi)),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(I5,vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),C,Q)
           => hoare_1429296392585015714_VEBTi(P,C,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_4246_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F4: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( entails(P,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(Uu),Uua)),Xi)),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(I5,vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),C,Q)
           => hoare_3067605981109127869le_nat(P,C,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_4247_snga__same__false,axiom,
    ! [P3: array_VEBT_VEBTi,X3: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(P3,X3)),snga_assn_VEBT_VEBTi(P3,Y)) = bot_bot_assn ).

% snga_same_false
tff(fact_4248_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_real,Y: vEBT_VEBT,X3: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_list_real_nat(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),X3)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X3)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,I),Y),list_u6098035379799741383_VEBTi(Tree_is,I,X3))),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,X3))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,I),Y)),list_u6098035379799741383_VEBTi(Tree_is,I,X3)))) ) ) ).

% tcd
tff(fact_4249_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_bool,Y: vEBT_VEBT,X3: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_list_bool_nat(size_size_list_bool,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),X3)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X3)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,I),Y),list_u6098035379799741383_VEBTi(Tree_is,I,X3))),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,X3))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,I),Y)),list_u6098035379799741383_VEBTi(Tree_is,I,X3)))) ) ) ).

% tcd
tff(fact_4250_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_nat,Y: vEBT_VEBT,X3: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),X3)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X3)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,I),Y),list_u6098035379799741383_VEBTi(Tree_is,I,X3))),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,X3))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,I),Y)),list_u6098035379799741383_VEBTi(Tree_is,I,X3)))) ) ) ).

% tcd
tff(fact_4251_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_int,Y: vEBT_VEBT,X3: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_list_int_nat(size_size_list_int,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),X3)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X3)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,I),Y),list_u6098035379799741383_VEBTi(Tree_is,I,X3))),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,X3))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,I),Y)),list_u6098035379799741383_VEBTi(Tree_is,I,X3)))) ) ) ).

% tcd
tff(fact_4252_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X3)),I) = X3 ) ) ).

% nth_list_update_eq
tff(fact_4253_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3)),I) = X3 ) ) ).

% nth_list_update_eq
tff(fact_4254_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
     => ( aa_nat_real(nth_real(aa_real_list_real(list_update_real(Xs,I),X3)),I) = X3 ) ) ).

% nth_list_update_eq
tff(fact_4255_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X3)),I))
      <=> pp(X3) ) ) ).

% nth_list_update_eq
tff(fact_4256_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X3)),I) = X3 ) ) ).

% nth_list_update_eq
tff(fact_4257_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X3)),I) = X3 ) ) ).

% nth_list_update_eq
tff(fact_4258_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J: nat,X3: vEBT_VEBTi] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,J,X3)),I) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I) ) ) ).

% nth_update_invalid
tff(fact_4259_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J: nat,X3: vEBT_VEBT] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,J),X3)),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I) ) ) ).

% nth_update_invalid
tff(fact_4260_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J: nat,X3: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
     => ( aa_nat_real(nth_real(aa_real_list_real(list_update_real(L,J),X3)),I) = aa_nat_real(nth_real(L),I) ) ) ).

% nth_update_invalid
tff(fact_4261_nth__update__invalid,axiom,
    ! [I: nat,L: list_bool,J: nat,X3: bool] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(L,J,X3)),I))
      <=> pp(aa_nat_bool(nth_bool(L),I)) ) ) ).

% nth_update_invalid
tff(fact_4262_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,X3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
     => ( aa_nat_nat(nth_nat(list_update_nat(L,J,X3)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_update_invalid
tff(fact_4263_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,X3: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,L)))
     => ( aa_nat_int(nth_int(list_update_int(L,J,X3)),I) = aa_nat_int(nth_int(L),I) ) ) ).

% nth_update_invalid
tff(fact_4264_atLeastLessThan__singleton,axiom,
    ! [M: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_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_4265_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X3: vEBT_VEBTi,Xb: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( Xaa = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),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),X3)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xb),aa_VEB7679222860772546456on_nat(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,H,X3))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList))),aa_set_nat_set_nat(insert_nat(H),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,H,X3))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xb),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Xaa))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),Xaa)),list_u6098035379799741383_VEBTi(Tree_is,H,X3)))) ) ) ).

% big_assn_simp'
tff(fact_4266_big__assn__simp,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,L: nat,X3: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,H),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),H)),L)),X3)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xaa),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),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,H,X3))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList))),aa_set_nat_set_nat(insert_nat(H),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,H,X3))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xaa),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L)))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),list_u6098035379799741383_VEBTi(Tree_is,H,X3)))) ) ).

% big_assn_simp
tff(fact_4267_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4268_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
       => ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(aa_VEB5888937730619358530T_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_4269_set__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_real_nat(size_size_list_real,Xs)))
       => ( aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(aa_real_list_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_4270_set__swap,axiom,
    ! [I: nat,Xs: list_bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(list_update_bool(Xs,I,aa_nat_bool(nth_bool(Xs),J)),J,aa_nat_bool(nth_bool(Xs),I))) = aa_lis3587183946281582166t_bool(set_bool2,Xs) ) ) ) ).

% set_swap
tff(fact_4271_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4272_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(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_4273_distinct__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_4274_distinct__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
       => ( distinct_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(aa_VEB5888937730619358530T_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)))
        <=> distinct_VEBT_VEBT(Xs) ) ) ) ).

% distinct_swap
tff(fact_4275_distinct__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_list_real_bool(distinct_real,aa_real_list_real(list_update_real(aa_real_list_real(list_update_real(Xs,I),aa_nat_real(nth_real(Xs),J)),J),aa_nat_real(nth_real(Xs),I))))
        <=> pp(aa_list_real_bool(distinct_real,Xs)) ) ) ) ).

% distinct_swap
tff(fact_4276_distinct__swap,axiom,
    ! [I: nat,Xs: list_bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( pp(aa_list_bool_bool(distinct_bool,list_update_bool(list_update_bool(Xs,I,aa_nat_bool(nth_bool(Xs),J)),J,aa_nat_bool(nth_bool(Xs),I))))
        <=> pp(aa_list_bool_bool(distinct_bool,Xs)) ) ) ) ).

% distinct_swap
tff(fact_4277_distinct__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_list_nat_bool(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))))
        <=> pp(aa_list_nat_bool(distinct_nat,Xs)) ) ) ) ).

% distinct_swap
tff(fact_4278_distinct__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_list_int_nat(size_size_list_int,Xs)))
       => ( pp(aa_list_int_bool(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))))
        <=> pp(aa_list_int_bool(distinct_int,Xs)) ) ) ) ).

% distinct_swap
tff(fact_4279_snga__prec,axiom,
    precis1518117471652397933_VEBTi(aTP_Lamm_os) ).

% snga_prec
tff(fact_4280_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_set_real(set_or66887138388493659n_real(A),B) = aa_real_set_real(set_or66887138388493659n_real(C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4281_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4282_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4283_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4284_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_set_int(set_or4662586982721622107an_int(A),B) = aa_int_set_int(set_or4662586982721622107an_int(C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4285_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C,D) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4286_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_set_real(set_or66887138388493659n_real(A),B) = aa_real_set_real(set_or66887138388493659n_real(C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4287_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4288_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4289_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4290_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_set_int(set_or4662586982721622107an_int(A),B) = aa_int_set_int(set_or4662586982721622107an_int(C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4291_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C,D) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4292_atLeastLessThan__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( ( aa_real_set_real(set_or66887138388493659n_real(A),B) = aa_real_set_real(set_or66887138388493659n_real(C),D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4293_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4294_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),D))
       => ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4295_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4296_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( ( aa_int_set_int(set_or4662586982721622107an_int(A),B) = aa_int_set_int(set_or4662586982721622107an_int(C),D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4297_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
       => ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4298_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X3: set_nat,I: nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A2))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,X3))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4299_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X3: nat,I: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Xs)),A2))
     => ( pp(aa_set_nat_bool(member_nat(X3),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,list_update_nat(Xs,I,X3))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4300_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X3: real,I: nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),A2))
     => ( pp(aa_set_real_bool(member_real(X3),A2))
       => pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(Xs,I),X3))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4301_set__update__subsetI,axiom,
    ! [Xs: list_bool,A2: set_bool,X3: bool,I: nat] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs)),A2))
     => ( pp(aa_set_bool_bool(member_bool(X3),A2))
       => pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(Xs,I,X3))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4302_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBTi,A2: set_VEBT_VEBTi,X3: vEBT_VEBTi,I: nat] :
      ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Xs),A2)
     => ( member_VEBT_VEBTi(X3,A2)
       => ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,X3)),A2) ) ) ).

% set_update_subsetI
tff(fact_4303_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X3: vEBT_VEBT,I: nat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),A2))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4304_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X3: int,I: nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Xs)),A2))
     => ( pp(aa_set_int_bool(member_int(X3),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,list_update_int(Xs,I,X3))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4305_infinite__Ico,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_set_real_bool(finite_finite_real,aa_real_set_real(set_or66887138388493659n_real(A),B))) ) ).

% infinite_Ico
tff(fact_4306_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_set_rat_bool(finite_finite_rat,set_or4029947393144176647an_rat(A,B))) ) ).

% infinite_Ico
tff(fact_4307_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A,B)),set_or4029947393144176647an_rat(C,D)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4308_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or1222409239386451017an_num(A,B)),set_or1222409239386451017an_num(C,D)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4309_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,C),D)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4310_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(set_or4662586982721622107an_int(A),B)),aa_int_set_int(set_or4662586982721622107an_int(C),D)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4311_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(A,B)),set_or8404916559141939852nteger(C,D)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4312_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
         => pp(aa_nat_bool(P,M5)) )
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X4)) ) ) ).

% all_nat_less_eq
tff(fact_4313_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
          & pp(aa_nat_bool(P,M5)) )
    <=> ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X4)) ) ) ).

% ex_nat_less_eq
tff(fact_4314_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_4315_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N7),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N7)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_4316_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_set_nat,X3: set_nat,Y: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3254054031482475050et_nat(L)))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(list_update_set_nat(L,I,Y))))
      <=> ( ( X3 = Y )
          | ! [Y5: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(list_update_set_nat(L,I,Y5)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4317_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X3: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( X3 = Y )
          | ! [Y5: vEBT_VEBTi] : member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4318_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X3: vEBT_VEBT,Y: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),Y))))
      <=> ( ( X3 = Y )
          | ! [Y5: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),Y5)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4319_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,X3: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
     => ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(L,I),Y))))
      <=> ( ( X3 = Y )
          | ! [Y5: real] : pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(L,I),Y5)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4320_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_bool,X3: bool,Y: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
     => ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(L,I,Y))))
      <=> ( ( pp(X3)
          <=> pp(Y) )
          | ! [Y5: bool] : pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(L,I,Y5)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4321_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
     => ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y))))
      <=> ( ( X3 = Y )
          | ! [Y5: nat] : pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y5)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4322_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,X3: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,L)))
     => ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,list_update_int(L,I,Y))))
      <=> ( ( X3 = Y )
          | ! [Y5: int] : pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,list_update_int(L,I,Y5)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4323_in__set__upd__cases,axiom,
    ! [X3: set_nat,L: list_set_nat,I: nat,Y: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(list_update_set_nat(L,I,Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3254054031482475050et_nat(L)))
         => ( X3 != Y ) )
       => pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(L))) ) ) ).

% in_set_upd_cases
tff(fact_4324_in__set__upd__cases,axiom,
    ! [X3: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
         => ( X3 != Y ) )
       => member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(L)) ) ) ).

% in_set_upd_cases
tff(fact_4325_in__set__upd__cases,axiom,
    ! [X3: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
         => ( X3 != Y ) )
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L))) ) ) ).

% in_set_upd_cases
tff(fact_4326_in__set__upd__cases,axiom,
    ! [X3: real,L: list_real,I: nat,Y: real] :
      ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(L,I),Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
         => ( X3 != Y ) )
       => pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,L))) ) ) ).

% in_set_upd_cases
tff(fact_4327_in__set__upd__cases,axiom,
    ! [X3: bool,L: list_bool,I: nat,Y: bool] :
      ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(L,I,Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
         => ( pp(X3)
          <=> ~ pp(Y) ) )
       => pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,L))) ) ) ).

% in_set_upd_cases
tff(fact_4328_in__set__upd__cases,axiom,
    ! [X3: nat,L: list_nat,I: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
         => ( X3 != Y ) )
       => pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,L))) ) ) ).

% in_set_upd_cases
tff(fact_4329_in__set__upd__cases,axiom,
    ! [X3: int,L: list_int,I: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,list_update_int(L,I,Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,L)))
         => ( X3 != Y ) )
       => pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,L))) ) ) ).

% in_set_upd_cases
tff(fact_4330_in__set__upd__eq,axiom,
    ! [I: nat,L: list_set_nat,X3: set_nat,Y: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3254054031482475050et_nat(L)))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(list_update_set_nat(L,I,Y))))
      <=> ( ( X3 = Y )
          | ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(L)))
            & ! [Y5: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(list_update_set_nat(L,I,Y5)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4331_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X3: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( X3 = Y )
          | ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(L))
            & ! [Y5: vEBT_VEBTi] : member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4332_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X3: vEBT_VEBT,Y: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),Y))))
      <=> ( ( X3 = Y )
          | ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L)))
            & ! [Y5: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),Y5)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4333_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,X3: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
     => ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(L,I),Y))))
      <=> ( ( X3 = Y )
          | ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,L)))
            & ! [Y5: real] : pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(L,I),Y5)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4334_in__set__upd__eq,axiom,
    ! [I: nat,L: list_bool,X3: bool,Y: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
     => ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(L,I,Y))))
      <=> ( ( pp(X3)
          <=> pp(Y) )
          | ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,L)))
            & ! [Y5: bool] : pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(L,I,Y5)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4335_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
     => ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y))))
      <=> ( ( X3 = Y )
          | ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,L)))
            & ! [Y5: nat] : pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y5)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4336_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,X3: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,L)))
     => ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,list_update_int(L,I,Y))))
      <=> ( ( X3 = Y )
          | ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,L)))
            & ! [Y5: int] : pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,list_update_int(L,I,Y5)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4337_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X3: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3254054031482475050et_nat(Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(list_update_set_nat(Xs,N,X3)))) ) ).

% set_update_memI
tff(fact_4338_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,N,X3))) ) ).

% set_update_memI
tff(fact_4339_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,N),X3)))) ) ).

% set_update_memI
tff(fact_4340_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
     => pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(Xs,N),X3)))) ) ).

% set_update_memI
tff(fact_4341_set__update__memI,axiom,
    ! [N: nat,Xs: list_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
     => pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(Xs,N,X3)))) ) ).

% set_update_memI
tff(fact_4342_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,list_update_nat(Xs,N,X3)))) ) ).

% set_update_memI
tff(fact_4343_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
     => pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,list_update_int(Xs,N,X3)))) ) ).

% set_update_memI
tff(fact_4344_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,list_update_nat(Xs,I,X3))),aa_set_nat_set_nat(insert_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))) ).

% set_update_subset_insert
tff(fact_4345_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X3: real] : pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(Xs,I),X3))),aa_set_real_set_real(insert_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))) ).

% set_update_subset_insert
tff(fact_4346_set__update__subset__insert,axiom,
    ! [Xs: list_bool,I: nat,X3: bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(Xs,I,X3))),aa_set_bool_set_bool(insert_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))) ).

% set_update_subset_insert
tff(fact_4347_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X3: vEBT_VEBTi] : ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,X3)),insert_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))) ).

% set_update_subset_insert
tff(fact_4348_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3))),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))) ).

% set_update_subset_insert
tff(fact_4349_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,list_update_int(Xs,I,X3))),aa_set_int_set_int(insert_int(X3),aa_list_int_set_int(set_int2,Xs)))) ).

% set_update_subset_insert
tff(fact_4350_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( ( list_u6098035379799741383_VEBTi(Xs,I,X3) = Xs )
      <=> ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I) = X3 ) ) ) ).

% list_update_same_conv
tff(fact_4351_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3) = Xs )
      <=> ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I) = X3 ) ) ) ).

% list_update_same_conv
tff(fact_4352_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
     => ( ( aa_real_list_real(list_update_real(Xs,I),X3) = Xs )
      <=> ( aa_nat_real(nth_real(Xs),I) = X3 ) ) ) ).

% list_update_same_conv
tff(fact_4353_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( ( list_update_bool(Xs,I,X3) = Xs )
      <=> ( pp(aa_nat_bool(nth_bool(Xs),I))
        <=> pp(X3) ) ) ) ).

% list_update_same_conv
tff(fact_4354_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ( list_update_nat(Xs,I,X3) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X3 ) ) ) ).

% list_update_same_conv
tff(fact_4355_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( ( list_update_int(Xs,I,X3) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = X3 ) ) ) ).

% list_update_same_conv
tff(fact_4356_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X3)),J) = X3 ) )
        & ( ( I != J )
         => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X3)),J) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_4357_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3)),J) = X3 ) )
        & ( ( I != J )
         => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3)),J) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_4358_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,J: nat,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
     => ( ( ( I = J )
         => ( aa_nat_real(nth_real(aa_real_list_real(list_update_real(Xs,I),X3)),J) = X3 ) )
        & ( ( I != J )
         => ( aa_nat_real(nth_real(aa_real_list_real(list_update_real(Xs,I),X3)),J) = aa_nat_real(nth_real(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_4359_nth__list__update,axiom,
    ! [I: nat,Xs: list_bool,X3: bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X3)),J))
      <=> ( ( ( I = J )
           => pp(X3) )
          & ( ( I != J )
           => pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ).

% nth_list_update
tff(fact_4360_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( ( ( I = J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X3)),J) = X3 ) )
        & ( ( I != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X3)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_4361_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( ( ( I = J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X3)),J) = X3 ) )
        & ( ( I != J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X3)),J) = aa_nat_int(nth_int(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_4362_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))) )
       => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,X3)),J) = X3 ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))) )
       => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,X3)),J) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),J) ) ) ) ).

% nth_list_update'
tff(fact_4363_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))) )
       => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),X3)),J) = X3 ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))) )
       => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),X3)),J) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),J) ) ) ) ).

% nth_list_update'
tff(fact_4364_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_real,X3: real] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L))) )
       => ( aa_nat_real(nth_real(aa_real_list_real(list_update_real(L,I),X3)),J) = X3 ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L))) )
       => ( aa_nat_real(nth_real(aa_real_list_real(list_update_real(L,I),X3)),J) = aa_nat_real(nth_real(L),J) ) ) ) ).

% nth_list_update'
tff(fact_4365_nth__list__update_H,axiom,
    ! [L: list_bool,I: nat,X3: bool,J: nat] :
      ( pp(aa_nat_bool(nth_bool(list_update_bool(L,I,X3)),J))
    <=> ( ( ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L))) )
         => pp(X3) )
        & ( ~ ( ( I = J )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L))) )
         => pp(aa_nat_bool(nth_bool(L),J)) ) ) ) ).

% nth_list_update'
tff(fact_4366_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_nat,X3: nat] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L))) )
       => ( aa_nat_nat(nth_nat(list_update_nat(L,I,X3)),J) = X3 ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L))) )
       => ( aa_nat_nat(nth_nat(list_update_nat(L,I,X3)),J) = aa_nat_nat(nth_nat(L),J) ) ) ) ).

% nth_list_update'
tff(fact_4367_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_int,X3: int] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,L))) )
       => ( aa_nat_int(nth_int(list_update_int(L,I,X3)),J) = X3 ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,L))) )
       => ( aa_nat_int(nth_int(list_update_int(L,I,X3)),J) = aa_nat_int(nth_int(L),J) ) ) ) ).

% nth_list_update'
tff(fact_4368_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE2561185598974572172i_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L886525131989349516_VEBTi(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4369_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5577048379444288083T_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2497118539674116125T_VEBT(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4370_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(I5,A2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4371_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L7851252805511451907_VEBTi(I5,A2,aa_real_list_real(list_update_real(Xs,I),X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4372_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(I5,A2,aa_real_list_real(list_update_real(Xs,I),X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4373_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_bool,A2: fun_bo7535889243153246212i_assn,X1: bool,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( vEBT_L6372884366951783724_VEBTi(I5,A2,list_update_bool(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L6372884366951783724_VEBTi(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4374_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_bool,A2: fun_bo5394993412183903963T_assn,X1: bool,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( vEBT_L2992182275864018813T_VEBT(I5,A2,list_update_bool(Xs,I,X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = vEBT_L2992182275864018813T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4375_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: fun_na6467670216208870567i_assn,X1: nat,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( vEBT_L7489483478785760935_VEBTi(I5,A2,list_update_nat(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L7489483478785760935_VEBTi(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4376_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: fun_na4489259242115493880T_assn,X1: nat,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( vEBT_L8511957252848910786T_VEBT(I5,A2,list_update_nat(Xs,I,X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = vEBT_L8511957252848910786T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4377_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A2: fun_in1903134756389476099i_assn,X1: int,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
       => ( vEBT_L114188773329725699_VEBTi(I5,A2,list_update_int(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L114188773329725699_VEBTi(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4378_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N)) = aa_set_nat_set_nat(insert_nat(N),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_4379_map__upd__eq,axiom,
    ! [I: nat,L: list_P2722043638750504429l_bool,F: fun_Pr4298194452408323251l_bool,X3: produc2804947525362035943l_bool] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6988337448923680857l_bool(L)))
       => ( pp(aa_Pro9114106320147817132l_bool(F,nth_Pr2856333732135533678l_bool(L,I)))
        <=> pp(aa_Pro9114106320147817132l_bool(F,X3)) ) )
     => ( map_Pr8574941368599290932l_bool(F,list_u9160741145793889621l_bool(L,I,X3)) = map_Pr8574941368599290932l_bool(F,L) ) ) ).

% map_upd_eq
tff(fact_4380_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,X3: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
       => ( aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_nat(F,X3) ) )
     => ( map_VEBT_VEBT_nat(F,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),X3)) = map_VEBT_VEBT_nat(F,L) ) ) ).

% map_upd_eq
tff(fact_4381_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,X3: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
       => ( aa_VEBT_VEBT_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_real(F,X3) ) )
     => ( map_VEBT_VEBT_real(F,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),X3)) = map_VEBT_VEBT_real(F,L) ) ) ).

% map_upd_eq
tff(fact_4382_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: fun_nat_nat,X3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
       => ( aa_nat_nat(F,aa_nat_nat(nth_nat(L),I)) = aa_nat_nat(F,X3) ) )
     => ( map_nat_nat(F,list_update_nat(L,I,X3)) = map_nat_nat(F,L) ) ) ).

% map_upd_eq
tff(fact_4383_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: fun_nat_bool,X3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
       => ( pp(aa_nat_bool(F,aa_nat_nat(nth_nat(L),I)))
        <=> pp(aa_nat_bool(F,X3)) ) )
     => ( map_nat_bool(F,list_update_nat(L,I,X3)) = map_nat_bool(F,L) ) ) ).

% map_upd_eq
tff(fact_4384_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N)) = aa_set_nat_set_nat(insert_nat(N),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_4385_listI__assn__conv,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A2: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi] :
      ( ( N = size_s6755466524823107622T_VEBT(Xs) )
     => ( vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N),A2,Xs,Xsi) = aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A2,Xs),Xsi) ) ) ).

% listI_assn_conv
tff(fact_4386_list__assn__conv__idx,axiom,
    ! [A2: fun_VE905575822219544865i_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A2,Xs),Xsi) = vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),A2,Xs,Xsi) ).

% list_assn_conv_idx
tff(fact_4387_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3))) = insert_VEBT_VEBTi(X3,set_VEBT_VEBTi2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_4388_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,I),X3))) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_4389_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
     => ( aa_set_real_set_real(insert_real(aa_nat_real(nth_real(L),I)),aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(L,I),X3))) = aa_set_real_set_real(insert_real(X3),aa_lis3769085826154824284t_real(set_real2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_4390_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
     => ( aa_set_bool_set_bool(insert_bool(aa_nat_bool(nth_bool(L),I)),aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(L,I,X3))) = aa_set_bool_set_bool(insert_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_4391_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3))) = aa_set_nat_set_nat(insert_nat(X3),aa_list_nat_set_nat(set_nat2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_4392_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(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,X3))) = aa_set_int_set_int(insert_int(X3),aa_list_int_set_int(set_int2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_4393_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE2561185598974572172i_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A2,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(A2,X1),X2)),vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4394_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5577048379444288083T_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A2,list_u6098035379799741383_VEBTi(Xs,I,X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,X1),X2)),vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4395_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,X1),X2)),vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4396_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A2,aa_real_list_real(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(A2,X1),X2)),vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4397_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A2,aa_real_list_real(list_update_real(Xs,I),X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,X1),X2)),vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4398_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_bool,A2: fun_bo7535889243153246212i_assn,X1: bool,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( vEBT_L6372884366951783724_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A2,list_update_bool(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(A2,X1),X2)),vEBT_L6372884366951783724_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4399_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_bool,A2: fun_bo5394993412183903963T_assn,X1: bool,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( vEBT_L2992182275864018813T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A2,list_update_bool(Xs,I,X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A2,X1),X2)),vEBT_L2992182275864018813T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4400_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: fun_na6467670216208870567i_assn,X1: nat,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),A2,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(A2,X1),X2)),vEBT_L7489483478785760935_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4401_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: fun_na4489259242115493880T_assn,X1: nat,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),A2,list_update_nat(Xs,I,X1),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A2,X1),X2)),vEBT_L8511957252848910786T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4402_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A2: fun_in1903134756389476099i_assn,X1: int,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
       => ( vEBT_L114188773329725699_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A2,list_update_int(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(A2,X1),X2)),vEBT_L114188773329725699_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_4403_listI__assn__conv_H,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A2: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi,F4: assn] :
      ( ( N = size_s6755466524823107622T_VEBT(Xs) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N),A2,Xs,Xsi)),F4) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A2,Xs),Xsi)),F4) ) ) ).

% listI_assn_conv'
tff(fact_4404_distinct__list__update,axiom,
    ! [Xs: list_set_nat,A: set_nat,I: nat] :
      ( distinct_set_nat(Xs)
     => ( ~ pp(aa_set_set_nat_bool(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_4405_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_4406_distinct__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,A: vEBT_VEBT,I: nat] :
      ( distinct_VEBT_VEBT(Xs)
     => ( ~ pp(aa_set8260762808118238015T_bool(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))))
       => distinct_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),A)) ) ) ).

% distinct_list_update
tff(fact_4407_distinct__list__update,axiom,
    ! [Xs: list_real,A: real,I: nat] :
      ( pp(aa_list_real_bool(distinct_real,Xs))
     => ( ~ pp(aa_set_real_bool(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))))
       => pp(aa_list_real_bool(distinct_real,aa_real_list_real(list_update_real(Xs,I),A))) ) ) ).

% distinct_list_update
tff(fact_4408_distinct__list__update,axiom,
    ! [Xs: list_bool,A: bool,I: nat] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
     => ( ~ pp(aa_set_bool_bool(member_bool(A),minus_minus_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs),aa_set_bool_set_bool(insert_bool(aa_nat_bool(nth_bool(Xs),I)),bot_bot_set_bool))))
       => pp(aa_list_bool_bool(distinct_bool,list_update_bool(Xs,I,A))) ) ) ).

% distinct_list_update
tff(fact_4409_distinct__list__update,axiom,
    ! [Xs: list_int,A: int,I: nat] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( ~ pp(aa_set_int_bool(member_int(A),aa_set_int_set_int(aa_set4234024024286713313et_int(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))))
       => pp(aa_list_int_bool(distinct_int,list_update_int(Xs,I,A))) ) ) ).

% distinct_list_update
tff(fact_4410_distinct__list__update,axiom,
    ! [Xs: list_nat,A: nat,I: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( ~ pp(aa_set_nat_bool(member_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(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))))
       => pp(aa_list_nat_bool(distinct_nat,list_update_nat(Xs,I,A))) ) ) ).

% distinct_list_update
tff(fact_4411_set__update__distinct,axiom,
    ! [Xs: list_VEBT_VEBTi,N: nat,X3: vEBT_VEBTi] :
      ( distinct_VEBT_VEBTi(Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
       => ( set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,N,X3)) = insert_VEBT_VEBTi(X3,minus_3697805406911847364_VEBTi(set_VEBT_VEBTi2(Xs),insert_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N),bot_bo8982466882572371071_VEBTi))) ) ) ) ).

% set_update_distinct
tff(fact_4412_set__update__distinct,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X3: vEBT_VEBT] :
      ( distinct_VEBT_VEBT(Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,N),X3)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),minus_5127226145743854075T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)),bot_bo8194388402131092736T_VEBT))) ) ) ) ).

% set_update_distinct
tff(fact_4413_set__update__distinct,axiom,
    ! [Xs: list_real,N: nat,X3: real] :
      ( pp(aa_list_real_bool(distinct_real,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
       => ( aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(list_update_real(Xs,N),X3)) = aa_set_real_set_real(insert_real(X3),minus_minus_set_real(aa_lis3769085826154824284t_real(set_real2,Xs),aa_set_real_set_real(insert_real(aa_nat_real(nth_real(Xs),N)),bot_bot_set_real))) ) ) ) ).

% set_update_distinct
tff(fact_4414_set__update__distinct,axiom,
    ! [Xs: list_bool,N: nat,X3: bool] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( aa_lis3587183946281582166t_bool(set_bool2,list_update_bool(Xs,N,X3)) = aa_set_bool_set_bool(insert_bool(X3),minus_minus_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs),aa_set_bool_set_bool(insert_bool(aa_nat_bool(nth_bool(Xs),N)),bot_bot_set_bool))) ) ) ) ).

% set_update_distinct
tff(fact_4415_set__update__distinct,axiom,
    ! [Xs: list_int,N: nat,X3: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
       => ( aa_list_int_set_int(set_int2,list_update_int(Xs,N,X3)) = aa_set_int_set_int(insert_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(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),N)),bot_bot_set_int))) ) ) ) ).

% set_update_distinct
tff(fact_4416_set__update__distinct,axiom,
    ! [Xs: list_nat,N: nat,X3: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( aa_list_nat_set_nat(set_nat2,list_update_nat(Xs,N,X3)) = aa_set_nat_set_nat(insert_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(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),N)),bot_bot_set_nat))) ) ) ) ).

% set_update_distinct
tff(fact_4417_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,X3: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F4: assn,Q: assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L886525131989349516_VEBTi(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4418_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,X3: vEBT_VEBTi,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F4: assn,Q: assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2497118539674116125T_VEBT(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4419_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,X3: vEBT_VEBT,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F4: assn,Q: assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3204528365124325536T_VEBT(I5,A2,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4420_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,X3: real,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F4: assn,Q: assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7851252805511451907_VEBTi(I5,A2,aa_real_list_real(list_update_real(Xs,I),X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4421_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,X3: real,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F4: assn,Q: assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3095048238742455910T_VEBT(I5,A2,aa_real_list_real(list_update_real(Xs,I),X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4422_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_bo7535889243153246212i_assn,X3: bool,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBTi,F4: assn,Q: assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(A2,X3),Xi)),vEBT_L6372884366951783724_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6372884366951783724_VEBTi(I5,A2,list_update_bool(Xs,I,X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4423_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_bo5394993412183903963T_assn,X3: bool,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBT,F4: assn,Q: assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A2,X3),Xi)),vEBT_L2992182275864018813T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2992182275864018813T_VEBT(I5,A2,list_update_bool(Xs,I,X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4424_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_na6467670216208870567i_assn,X3: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F4: assn,Q: assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L7489483478785760935_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7489483478785760935_VEBTi(I5,A2,list_update_nat(Xs,I,X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4425_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_na4489259242115493880T_assn,X3: nat,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F4: assn,Q: assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L8511957252848910786T_VEBT(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8511957252848910786T_VEBT(I5,A2,list_update_nat(Xs,I,X3),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xsi,I),Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4426_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_in1903134756389476099i_assn,X3: int,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_int,Xsi: list_VEBT_VEBTi,F4: assn,Q: assn] :
      ( entails(P,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(A2,X3),Xi)),vEBT_L114188773329725699_VEBTi(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A2,Xs,Xsi))),F4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L114188773329725699_VEBTi(I5,A2,list_update_int(Xs,I,X3),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_4427_nth__rule,axiom,
    ! [I: nat,Xs: list_int,A: array_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => hoare_3065115510600077593le_int(snga_assn_int(A,Xs),array_nth_int(A,I),aa_arr1589534196239669886t_assn(aa_lis459371148865260371t_assn(aTP_Lamm_ot(I),Xs),A)) ) ).

% nth_rule
tff(fact_4428_nth__rule,axiom,
    ! [I: nat,Xs: list_option_nat,A: array_option_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6086282163384603972on_nat(Xs)))
     => hoare_7629718768684598413on_nat(snga_assn_option_nat(A,Xs),array_nth_option_nat(A,I),aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_ou(I),Xs),A)) ) ).

% nth_rule
tff(fact_4429_nth__rule,axiom,
    ! [I: nat,Xs: list_bool,A: array_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
     => hoare_2774773592644289642e_bool(snga_assn_bool(A,Xs),array_nth_bool(A,I),aa_arr3483125744708554596l_assn(aa_lis2577335236791371354l_assn(aTP_Lamm_ov(I),Xs),A)) ) ).

% nth_rule
tff(fact_4430_nth__rule,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A: array_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => hoare_1429296392585015714_VEBTi(snga_assn_VEBT_VEBTi(A,Xs),array_nth_VEBT_VEBTi(A,I),aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_ow(I),Xs),A)) ) ).

% nth_rule
tff(fact_4431_nth__rule,axiom,
    ! [I: nat,Xs: list_nat,A: array_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => hoare_3067605981109127869le_nat(snga_assn_nat(A,Xs),array_nth_nat(A,I),aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_ox(I),Xs),A)) ) ).

% nth_rule
tff(fact_4432_freeze__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs: list_VEBT_VEBTi] : hoare_3904069481286416050_VEBTi(snga_assn_VEBT_VEBTi(A,Xs),array_8141364883501958055_VEBTi(A),aa_lis3169797078229555565i_assn(aTP_Lamm_oy(A),Xs)) ).

% freeze_rule
tff(fact_4433_upd__rule,axiom,
    ! [I: nat,Xs: list_bool,A: array_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
     => hoare_9097894755699397917y_bool(snga_assn_bool(A,Xs),array_upd_bool(I,X3,A),aa_boo8717665131543412656l_assn(aa_arr1850246612958046993l_assn(aa_lis1773218934604694151l_assn(aTP_Lamm_oz(I),Xs),A),X3)) ) ).

% upd_rule
tff(fact_4434_upd__rule,axiom,
    ! [I: nat,Xs: list_nat,A: array_nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => hoare_6807272225193264096ay_nat(snga_assn_nat(A,Xs),array_upd_nat(I,X3,A),aa_nat8392338049721622586t_assn(aa_arr4802002299793635008t_assn(aa_lis5007279772662053221t_assn(aTP_Lamm_pa(I),Xs),A),X3)) ) ).

% upd_rule
tff(fact_4435_upd__rule,axiom,
    ! [I: nat,Xs: list_int,A: array_int,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => hoare_2629421205684067388ay_int(snga_assn_int(A,Xs),array_upd_int(I,X3,A),aa_int2412489125148462906t_assn(aa_arr8066481932369824868t_assn(aa_lis2681267339321482597t_assn(aTP_Lamm_pb(I),Xs),A),X3)) ) ).

% upd_rule
tff(fact_4436_upd__rule,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A: array_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => hoare_3353465787467722821_VEBTi(snga_assn_VEBT_VEBTi(A,Xs),array_upd_VEBT_VEBTi(I,X3,A),aa_VEB7841561841333234298i_assn(aa_arr6331738488200487899i_assn(aa_lis1967528460011584869i_assn(aTP_Lamm_pc(I),Xs),A),X3)) ) ).

% upd_rule
tff(fact_4437_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi] : vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s7982070591426661849_VEBTi(Xsi)),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)))))),finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_lis8873771691224567519n_assn(aTP_Lamm_pd(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4438_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT] : vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s6755466524823107622T_VEBT(Xsi)),size_s7982070591426661849_VEBTi(Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)))))),finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_lis8759289953362995748n_assn(aTP_Lamm_pe(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4439_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT] : vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s6755466524823107622T_VEBT(Xsi)),size_s6755466524823107622T_VEBT(Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)))))),finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_lis4214038909079116777n_assn(aTP_Lamm_pf(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4440_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_re560840401387337987i_assn,Xs: list_real,Xsi: list_VEBT_VEBTi] : vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s7982070591426661849_VEBTi(Xsi)),aa_list_real_nat(size_size_list_real,Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs)))))),finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_lis6806030848160413398n_assn(aTP_Lamm_pg(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4441_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_re8050108611700907676T_assn,Xs: list_real,Xsi: list_VEBT_VEBT] : vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s6755466524823107622T_VEBT(Xsi)),aa_list_real_nat(size_size_list_real,Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs)))))),finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_lis7292822999309854957n_assn(aTP_Lamm_ph(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4442_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_bo7535889243153246212i_assn,Xs: list_bool,Xsi: list_VEBT_VEBTi] : vEBT_L6372884366951783724_VEBTi(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s7982070591426661849_VEBTi(Xsi)),aa_list_bool_nat(size_size_list_bool,Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_bool_nat(size_size_list_bool,Xs)))))),finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_lis2049715178687361725n_assn(aTP_Lamm_pi(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4443_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_bo5394993412183903963T_assn,Xs: list_bool,Xsi: list_VEBT_VEBT] : vEBT_L2992182275864018813T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s6755466524823107622T_VEBT(Xsi)),aa_list_bool_nat(size_size_list_bool,Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_bool_nat(size_size_list_bool,Xs)))))),finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_lis3096371939905232838n_assn(aTP_Lamm_pj(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4444_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_na6467670216208870567i_assn,Xs: list_nat,Xsi: list_VEBT_VEBTi] : vEBT_L7489483478785760935_VEBTi(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s7982070591426661849_VEBTi(Xsi)),aa_list_nat_nat(size_size_list_nat,Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs)))))),finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_lis1134338210885690618n_assn(aTP_Lamm_pk(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4445_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_na4489259242115493880T_assn,Xs: list_nat,Xsi: list_VEBT_VEBT] : vEBT_L8511957252848910786T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s6755466524823107622T_VEBT(Xsi)),aa_list_nat_nat(size_size_list_nat,Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs)))))),finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_lis3837078307170607049n_assn(aTP_Lamm_pl(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4446_listI__assn__def,axiom,
    ! [I5: set_nat,A2: fun_in1903134756389476099i_assn,Xs: list_int,Xsi: list_VEBT_VEBTi] : vEBT_L114188773329725699_VEBTi(I5,A2,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,size_s7982070591426661849_VEBTi(Xsi)),aa_list_int_nat(size_size_list_int,Xs))),aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,I5),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_int_nat(size_size_list_int,Xs)))))),finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_lis1861013024967981910n_assn(aTP_Lamm_pm(A2),Xs),Xsi),one_one_assn,I5)) ).

% listI_assn_def
tff(fact_4447_length__rule,axiom,
    ! [A: array_bool,Xs: list_bool] : hoare_3067605981109127869le_nat(snga_assn_bool(A,Xs),array_len_bool(A),aa_lis1286645363786127574t_assn(aTP_Lamm_pn(A),Xs)) ).

% length_rule
tff(fact_4448_length__rule,axiom,
    ! [A: array_nat,Xs: list_nat] : hoare_3067605981109127869le_nat(snga_assn_nat(A,Xs),array_len_nat(A),aa_lis6041139530851742867t_assn(aTP_Lamm_po(A),Xs)) ).

% length_rule
tff(fact_4449_length__rule,axiom,
    ! [A: array_int,Xs: list_int] : hoare_3067605981109127869le_nat(snga_assn_int(A,Xs),array_len_int(A),aa_lis2816288184639765743t_assn(aTP_Lamm_pp(A),Xs)) ).

% length_rule
tff(fact_4450_length__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs: list_VEBT_VEBTi] : hoare_3067605981109127869le_nat(snga_assn_VEBT_VEBTi(A,Xs),array_len_VEBT_VEBTi(A),aa_lis7348694692495225848t_assn(aTP_Lamm_pq(A),Xs)) ).

% length_rule
tff(fact_4451_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( distinct_VEBT_VEBT(L)
       => ( list_update_nat(map_VEBT_VEBT_nat(F,L),I,X3) = map_VEBT_VEBT_nat(fun_up6512855943550542919BT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4452_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( distinct_VEBT_VEBT(L)
       => ( aa_real_list_real(list_update_real(map_VEBT_VEBT_real(F,L),I),X3) = map_VEBT_VEBT_real(fun_up7749720967766031267T_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4453_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,F: fun_VE8449866691444886503_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( distinct_VEBT_VEBTi(L)
       => ( list_u6098035379799741383_VEBTi(map_VE483055756984248624_VEBTi(F,L),I,X3) = map_VE483055756984248624_VEBTi(fun_up8301472745780153321_VEBTi(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4454_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VE712148701811721852_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( distinct_VEBT_VEBT(L)
       => ( list_u6098035379799741383_VEBTi(map_VE7029150624388687525_VEBTi(F,L),I,X3) = map_VE7029150624388687525_VEBTi(fun_up377479910058727724_VEBTi(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4455_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,F: fun_VE9120655849212850072T_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( distinct_VEBT_VEBTi(L)
       => ( aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(map_VE7998069337340375161T_VEBT(F,L),I),X3) = map_VE7998069337340375161T_VEBT(fun_up1346398623010415360T_VEBT(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4456_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( distinct_VEBT_VEBT(L)
       => ( aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(map_VE8901447254227204932T_VEBT(F,L),I),X3) = map_VE8901447254227204932T_VEBT(fun_up224749957652071293T_VEBT(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4457_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_real,F: fun_real_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
     => ( pp(aa_list_real_bool(distinct_real,L))
       => ( list_u6098035379799741383_VEBTi(map_real_VEBT_VEBTi(F,L),I,X3) = map_real_VEBT_VEBTi(fun_up8049757234741329632_VEBTi(F,aa_nat_real(nth_real(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4458_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_real,F: fun_real_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
     => ( pp(aa_list_real_bool(distinct_real,L))
       => ( aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(map_real_VEBT_VEBT(F,L),I),X3) = map_real_VEBT_VEBT(fun_up6563732700392937161T_VEBT(F,aa_nat_real(nth_real(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4459_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_bool,F: fun_bool_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
     => ( pp(aa_list_bool_bool(distinct_bool,L))
       => ( list_u6098035379799741383_VEBTi(map_bool_VEBT_VEBTi(F,L),I,X3) = map_bool_VEBT_VEBTi(fun_up1662709444617513999_VEBTi(F,aa_nat_bool(nth_bool(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4460_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_bool,F: fun_bool_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
     => ( pp(aa_list_bool_bool(distinct_bool,L))
       => ( aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(map_bool_VEBT_VEBT(F,L),I),X3) = map_bool_VEBT_VEBT(fun_up4258218503615997530T_VEBT(F,aa_nat_bool(nth_bool(L),I),X3),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4461_horner__sum__foldr,axiom,
    ! [F: fun_real_real,A: real,Xs: list_real] : groups1503878375050959669l_real(F,A,Xs) = aa_real_real(foldr_real_real(aa_rea4187809805863855336l_real(aTP_Lamm_pr(F),A),Xs),zero_zero_real) ).

% horner_sum_foldr
tff(fact_4462_horner__sum__foldr,axiom,
    ! [F: fun_nat_nat,A: nat,Xs: list_nat] : aa_list_nat_nat(groups7488368174851004413at_nat(F,A),Xs) = aa_nat_nat(foldr_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ps(F),A),Xs),zero_zero_nat) ).

% horner_sum_foldr
tff(fact_4463_horner__sum__foldr,axiom,
    ! [F: fun_bool_int,A: int,Xs: list_bool] : groups4048646594741130792ol_int(F,A,Xs) = aa_int_int(foldr_bool_int(aa_int5463180763086053397nt_int(aTP_Lamm_pt(F),A),Xs),zero_zero_int) ).

% horner_sum_foldr
tff(fact_4464_set__remove1__eq,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( distinct_VEBT_VEBT(Xs)
     => ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,remove1_VEBT_VEBT(X3,Xs)) = minus_5127226145743854075T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)) ) ) ).

% set_remove1_eq
tff(fact_4465_set__remove1__eq,axiom,
    ! [Xs: list_real,X3: real] :
      ( pp(aa_list_real_bool(distinct_real,Xs))
     => ( aa_lis3769085826154824284t_real(set_real2,remove1_real(X3,Xs)) = minus_minus_set_real(aa_lis3769085826154824284t_real(set_real2,Xs),aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) ) ) ).

% set_remove1_eq
tff(fact_4466_set__remove1__eq,axiom,
    ! [Xs: list_bool,X3: bool] :
      ( pp(aa_list_bool_bool(distinct_bool,Xs))
     => ( aa_lis3587183946281582166t_bool(set_bool2,remove1_bool(X3,Xs)) = minus_minus_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs),aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)) ) ) ).

% set_remove1_eq
tff(fact_4467_set__remove1__eq,axiom,
    ! [Xs: list_int,X3: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( aa_list_int_set_int(set_int2,remove1_int(X3,Xs)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_list_int_set_int(set_int2,Xs)),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) ) ) ).

% set_remove1_eq
tff(fact_4468_set__remove1__eq,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( aa_list_nat_set_nat(set_nat2,remove1_nat(X3,Xs)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_list_nat_set_nat(set_nat2,Xs)),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) ) ) ).

% set_remove1_eq
tff(fact_4469_listsum__bound,axiom,
    ! [Xs: list_int,F: fun_int_real,Y: real] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_4470_listsum__bound,axiom,
    ! [Xs: list_set_nat,F: fun_set_nat_real,Y: real] :
      ( ! [X: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X),set_set_nat2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_set_nat_real(F,X))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_set_nat_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_4471_listsum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Y: real] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_4472_listsum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Y: real] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_4473_listsum__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,Y: real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_4474_listsum__bound,axiom,
    ! [Xs: list_bool,F: fun_bool_real,Y: real] :
      ( ! [X: bool] :
          ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(F,X))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_bool_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_4475_f__g__map__foldr__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,C: real,G: fun_VEBT_VEBT_real,D: real] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_VEBT_VEBT_real(G,X)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(G,Xs)),zero_zero_real))),D))) ) ).

% f_g_map_foldr_bound
tff(fact_4476_f__g__map__foldr__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,C: real,G: fun_nat_real,D: real] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_nat_real(G,X)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(G,Xs)),zero_zero_real))),D))) ) ).

% f_g_map_foldr_bound
tff(fact_4477_f__g__map__foldr__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,C: real,G: fun_real_real,D: real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(G,X)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(G,Xs)),zero_zero_real))),D))) ) ).

% f_g_map_foldr_bound
tff(fact_4478_f__g__map__foldr__bound,axiom,
    ! [Xs: list_bool,F: fun_bool_real,C: real,G: fun_bool_real,D: real] :
      ( ! [X: bool] :
          ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(F,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_bool_real(G,X)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_bool_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_bool_real(G,Xs)),zero_zero_real))),D))) ) ).

% f_g_map_foldr_bound
tff(fact_4479_set__remove1__subset,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,remove1_VEBT_VEBT(X3,Xs))),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))) ).

% set_remove1_subset
tff(fact_4480_set__remove1__subset,axiom,
    ! [X3: nat,Xs: list_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,remove1_nat(X3,Xs))),aa_list_nat_set_nat(set_nat2,Xs))) ).

% set_remove1_subset
tff(fact_4481_set__remove1__subset,axiom,
    ! [X3: real,Xs: list_real] : pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,remove1_real(X3,Xs))),aa_lis3769085826154824284t_real(set_real2,Xs))) ).

% set_remove1_subset
tff(fact_4482_set__remove1__subset,axiom,
    ! [X3: bool,Xs: list_bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,remove1_bool(X3,Xs))),aa_lis3587183946281582166t_bool(set_bool2,Xs))) ).

% set_remove1_subset
tff(fact_4483_set__remove1__subset,axiom,
    ! [X3: int,Xs: list_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,remove1_int(X3,Xs))),aa_list_int_set_int(set_int2,Xs))) ).

% set_remove1_subset
tff(fact_4484_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_4485_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_4486_sorted__remove1,axiom,
    ! [Xs: list_nat,A: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),remove1_nat(A,Xs))) ) ).

% sorted_remove1
tff(fact_4487_sorted__remove1,axiom,
    ! [Xs: list_int,A: int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),remove1_int(A,Xs))) ) ).

% sorted_remove1
tff(fact_4488_sorted__map__remove1,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),map_VEBT_VEBT_real(F,Xs)))
     => pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),map_VEBT_VEBT_real(F,remove1_VEBT_VEBT(X3,Xs)))) ) ).

% sorted_map_remove1
tff(fact_4489_sorted__map__remove1,axiom,
    ! [F: fun_Pr4298194452408323251l_bool,Xs: list_P2722043638750504429l_bool,X3: produc2804947525362035943l_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),map_Pr8574941368599290932l_bool(F,Xs)))
     => pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),map_Pr8574941368599290932l_bool(F,remove4234689833455435907l_bool(X3,Xs)))) ) ).

% sorted_map_remove1
tff(fact_4490_sorted__map__remove1,axiom,
    ! [F: fun_nat_bool,Xs: list_nat,X3: nat] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),map_nat_bool(F,Xs)))
     => pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),map_nat_bool(F,remove1_nat(X3,Xs)))) ) ).

% sorted_map_remove1
tff(fact_4491_sorted__map__remove1,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_VEBT_VEBT_nat(F,Xs)))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_VEBT_VEBT_nat(F,remove1_VEBT_VEBT(X3,Xs)))) ) ).

% sorted_map_remove1
tff(fact_4492_sorted__map__remove1,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,X3: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_nat_nat(F,Xs)))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_nat_nat(F,remove1_nat(X3,Xs)))) ) ).

% sorted_map_remove1
tff(fact_4493_length__remove1,axiom,
    ! [X3: set_nat,Xs: list_set_nat] :
      ( ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
       => ( size_s3254054031482475050et_nat(remove1_set_nat(X3,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s3254054031482475050et_nat(Xs)),one_one_nat) ) )
      & ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
       => ( size_s3254054031482475050et_nat(remove1_set_nat(X3,Xs)) = size_s3254054031482475050et_nat(Xs) ) ) ) ).

% length_remove1
tff(fact_4494_length__remove1,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
       => ( size_s6755466524823107622T_VEBT(remove1_VEBT_VEBT(X3,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
       => ( size_s6755466524823107622T_VEBT(remove1_VEBT_VEBT(X3,Xs)) = size_s6755466524823107622T_VEBT(Xs) ) ) ) ).

% length_remove1
tff(fact_4495_length__remove1,axiom,
    ! [X3: real,Xs: list_real] :
      ( ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
       => ( aa_list_real_nat(size_size_list_real,remove1_real(X3,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_real_nat(size_size_list_real,Xs)),one_one_nat) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
       => ( aa_list_real_nat(size_size_list_real,remove1_real(X3,Xs)) = aa_list_real_nat(size_size_list_real,Xs) ) ) ) ).

% length_remove1
tff(fact_4496_length__remove1,axiom,
    ! [X3: bool,Xs: list_bool] :
      ( ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
       => ( aa_list_bool_nat(size_size_list_bool,remove1_bool(X3,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_bool_nat(size_size_list_bool,Xs)),one_one_nat) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
       => ( aa_list_bool_nat(size_size_list_bool,remove1_bool(X3,Xs)) = aa_list_bool_nat(size_size_list_bool,Xs) ) ) ) ).

% length_remove1
tff(fact_4497_length__remove1,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
       => ( aa_list_nat_nat(size_size_list_nat,remove1_nat(X3,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
       => ( aa_list_nat_nat(size_size_list_nat,remove1_nat(X3,Xs)) = aa_list_nat_nat(size_size_list_nat,Xs) ) ) ) ).

% length_remove1
tff(fact_4498_length__remove1,axiom,
    ! [X3: int,Xs: list_int] :
      ( ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
       => ( aa_list_int_nat(size_size_list_int,remove1_int(X3,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_int_nat(size_size_list_int,Xs)),one_one_nat) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
       => ( aa_list_int_nat(size_size_list_int,remove1_int(X3,Xs)) = aa_list_int_nat(size_size_list_int,Xs) ) ) ) ).

% length_remove1
tff(fact_4499_real__nat__list,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(aTP_Lamm_pu(F),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_4500_real__nat__list,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(aTP_Lamm_pv(F),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_4501_new__rule,axiom,
    ! [N: nat,X3: bool] : hoare_9097894755699397917y_bool(one_one_assn,array_new_bool(N,X3),aa_boo8717665131543412656l_assn(aTP_Lamm_pw(N),X3)) ).

% new_rule
tff(fact_4502_new__rule,axiom,
    ! [N: nat,X3: vEBT_VEBTi] : hoare_3353465787467722821_VEBTi(one_one_assn,array_new_VEBT_VEBTi(N,X3),aa_VEB7841561841333234298i_assn(aTP_Lamm_px(N),X3)) ).

% new_rule
tff(fact_4503_aux,axiom,
    ! [P: fun_VE2561185598974572172i_assn,A: vEBT_VEBTi,As: list_VEBT_VEBTi,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] : finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_VEB7059816536500214223n_assn(aa_lis4146167562466521361n_assn(aa_VEB5563403434209864643n_assn(aTP_Lamm_py(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,size_s7982070591426661849_VEBTi(As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(P,A),C)),finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_lis8873771691224567519n_assn(aTP_Lamm_pd(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(As)))) ).

% aux
tff(fact_4504_aux,axiom,
    ! [P: fun_VE5577048379444288083T_assn,A: vEBT_VEBTi,As: list_VEBT_VEBTi,C: vEBT_VEBT,Cs: list_VEBT_VEBT] : finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_VEB3810757911076339055n_assn(aa_lis1293208971169636985n_assn(aa_VEB7709549983103476651n_assn(aTP_Lamm_pz(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,size_s7982070591426661849_VEBTi(As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(P,A),C)),finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_lis8759289953362995748n_assn(aTP_Lamm_pe(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(As)))) ).

% aux
tff(fact_4505_aux,axiom,
    ! [P: fun_VE905575822219544865i_assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] : finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_VEB7059816536500214223n_assn(aa_lis7703346581310142988n_assn(aa_VEB8713780084012957539n_assn(aTP_Lamm_qa(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,size_s6755466524823107622T_VEBT(As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P,A),C)),finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_lis4092487795791266010n_assn(aTP_Lamm_qb(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(As)))) ).

% aux
tff(fact_4506_aux,axiom,
    ! [P: fun_VE1571388752190420990T_assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: vEBT_VEBT,Cs: list_VEBT_VEBT] : finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_VEB3810757911076339055n_assn(aa_lis6098897284473852916n_assn(aa_VEB7190415443829630795n_assn(aTP_Lamm_qc(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,size_s6755466524823107622T_VEBT(As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(P,A),C)),finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_lis4214038909079116777n_assn(aTP_Lamm_pf(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(As)))) ).

% aux
tff(fact_4507_aux,axiom,
    ! [P: fun_VE1225457145571040039t_assn,A: vEBT_VEBTi,As: list_VEBT_VEBTi,C: nat,Cs: list_nat] : finite_fold_nat_assn(aa_lis8171252396799902776n_assn(aa_nat394924699201469189n_assn(aa_lis8390372872860913351n_assn(aa_VEB1074492319160947833n_assn(aTP_Lamm_qd(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,size_s7982070591426661849_VEBTi(As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(P,A),C)),finite_fold_nat_assn(aa_lis8171252396799902776n_assn(aa_lis2805659845025431162n_assn(aTP_Lamm_qe(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(As)))) ).

% aux
tff(fact_4508_aux,axiom,
    ! [P: fun_VE4931415397359990460t_assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: nat,Cs: list_nat] : finite_fold_nat_assn(aa_lis8171252396799902776n_assn(aa_nat394924699201469189n_assn(aa_lis8752055264758851010n_assn(aa_VEB7545035127168460825n_assn(aTP_Lamm_qf(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,size_s6755466524823107622T_VEBT(As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(P,A),C)),finite_fold_nat_assn(aa_lis8171252396799902776n_assn(aa_lis7929737622710957941n_assn(aTP_Lamm_qg(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(As)))) ).

% aux
tff(fact_4509_aux,axiom,
    ! [P: fun_VE5641251438918192003t_assn,A: vEBT_VEBTi,As: list_VEBT_VEBTi,C: int,Cs: list_int] : finite_fold_nat_assn(aa_lis157064186919824532n_assn(aa_int5700688002552083389n_assn(aa_lis8289800597781125503n_assn(aa_VEB219408230628803377n_assn(aTP_Lamm_qh(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,size_s7982070591426661849_VEBTi(As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(P,A),C)),finite_fold_nat_assn(aa_lis157064186919824532n_assn(aa_lis4014843672000128726n_assn(aTP_Lamm_qi(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(As)))) ).

% aux
tff(fact_4510_aux,axiom,
    ! [P: fun_VE123837653852366616t_assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: int,Cs: list_int] : finite_fold_nat_assn(aa_lis157064186919824532n_assn(aa_int5700688002552083389n_assn(aa_lis8651482989679063162n_assn(aa_VEB6689951038636316369n_assn(aTP_Lamm_qj(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,size_s6755466524823107622T_VEBT(As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB392321844297299145t_assn(P,A),C)),finite_fold_nat_assn(aa_lis157064186919824532n_assn(aa_lis9138921449685655505n_assn(aTP_Lamm_qk(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(As)))) ).

% aux
tff(fact_4511_aux,axiom,
    ! [P: fun_re560840401387337987i_assn,A: real,As: list_real,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] : finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_VEB7059816536500214223n_assn(aa_lis9048447917024140296n_assn(aa_rea7151089224645626673n_assn(aTP_Lamm_ql(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,aa_list_real_nat(size_size_list_real,As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(P,A),C)),finite_fold_nat_assn(aa_lis5217563234146327453n_assn(aa_lis6806030848160413398n_assn(aTP_Lamm_pg(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,As)))) ).

% aux
tff(fact_4512_aux,axiom,
    ! [P: fun_re8050108611700907676T_assn,A: real,As: list_real,C: vEBT_VEBT,Cs: list_VEBT_VEBT] : finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_VEB3810757911076339055n_assn(aa_lis7268569422076391408n_assn(aa_rea588685846782134297n_assn(aTP_Lamm_qm(P),A),As),C),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,aa_list_real_nat(size_size_list_real,As)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(P,A),C)),finite_fold_nat_assn(aa_lis2699940410532524056n_assn(aa_lis7292822999309854957n_assn(aTP_Lamm_ph(P),As),Cs),one_one_assn,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,As)))) ).

% aux
tff(fact_4513_of__list__rule,axiom,
    ! [Xs: list_VEBT_VEBTi] : hoare_3353465787467722821_VEBTi(one_one_assn,array_615059503499738533_VEBTi(Xs),aa_lis341327733535136106i_assn(aTP_Lamm_os,Xs)) ).

% of_list_rule
tff(fact_4514_foldr0,axiom,
    ! [Xs: list_real,C: real,D: real] : aa_real_real(foldr_real_real(plus_plus_real,Xs),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs),D)),C) ).

% foldr0
tff(fact_4515_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_4516_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_4517_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_4518_foldr__same,axiom,
    ! [Xs: list_real,Y: real] :
      ( ! [X: real,Y3: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => ( pp(aa_set_real_bool(member_real(Y3),aa_lis3769085826154824284t_real(set_real2,Xs)))
           => ( X = Y3 ) ) )
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(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,aa_list_real_nat(size_size_list_real,Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_4519_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Bound: real,I: real] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_s6755466524823107622T_VEBT(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_4520_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_real,F: fun_real_real,Bound: real,I: real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_list_real_nat(size_size_list_real,Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_4521_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_bool,F: fun_bool_real,Bound: real,I: real] :
      ( ! [X: bool] :
          ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(F,X)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_bool_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_list_bool_nat(size_size_list_bool,Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_4522_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Bound: real,I: real] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_list_nat_nat(size_size_list_nat,Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_4523_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_int,F: fun_int_real,Bound: real,I: real] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_list_int_nat(size_size_list_int,Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_4524_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_4525_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_4526_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_4527_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_4528_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_4529_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_4530_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_4531_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_4532_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_4533_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_4534_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% semiring_1_class.of_nat_0
tff(fact_4535_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_4536_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_4537_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_4538_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_4539_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_4540_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_4541_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_4542_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_4543_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_4544_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_4545_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_4546_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_4547_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_4548_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_4549_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_4550_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_4551_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_4552_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_4553_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_4554_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_4555_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_4556_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_4557_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_4558_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_4559_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_4560_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_4561_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_4562_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_4563_nth__Cons__Suc,axiom,
    ! [X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi,N: nat] : aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(X3,Xs)),aa_nat_nat(suc,N)) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N) ).

% nth_Cons_Suc
tff(fact_4564_nth__Cons__Suc,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] : aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs)),aa_nat_nat(suc,N)) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N) ).

% nth_Cons_Suc
tff(fact_4565_nth__Cons__Suc,axiom,
    ! [X3: nat,Xs: list_nat,N: nat] : aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(X3),Xs)),aa_nat_nat(suc,N)) = aa_nat_nat(nth_nat(Xs),N) ).

% nth_Cons_Suc
tff(fact_4566_nth__Cons__Suc,axiom,
    ! [X3: int,Xs: list_int,N: nat] : aa_nat_int(nth_int(aa_list_int_list_int(cons_int(X3),Xs)),aa_nat_nat(suc,N)) = aa_nat_int(nth_int(Xs),N) ).

% nth_Cons_Suc
tff(fact_4567_length__nth__simps_I4_J,axiom,
    ! [X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi,N: nat] : aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(X3,Xs)),aa_nat_nat(suc,N)) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N) ).

% length_nth_simps(4)
tff(fact_4568_length__nth__simps_I4_J,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] : aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs)),aa_nat_nat(suc,N)) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N) ).

% length_nth_simps(4)
tff(fact_4569_length__nth__simps_I4_J,axiom,
    ! [X3: nat,Xs: list_nat,N: nat] : aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(X3),Xs)),aa_nat_nat(suc,N)) = aa_nat_nat(nth_nat(Xs),N) ).

% length_nth_simps(4)
tff(fact_4570_length__nth__simps_I4_J,axiom,
    ! [X3: int,Xs: list_int,N: nat] : aa_nat_int(nth_int(aa_list_int_list_int(cons_int(X3),Xs)),aa_nat_nat(suc,N)) = aa_nat_int(nth_int(Xs),N) ).

% length_nth_simps(4)
tff(fact_4571_nth__Cons__0,axiom,
    ! [X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi] : aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(X3,Xs)),zero_zero_nat) = X3 ).

% nth_Cons_0
tff(fact_4572_nth__Cons__0,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] : aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs)),zero_zero_nat) = X3 ).

% nth_Cons_0
tff(fact_4573_nth__Cons__0,axiom,
    ! [X3: nat,Xs: list_nat] : aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(X3),Xs)),zero_zero_nat) = X3 ).

% nth_Cons_0
tff(fact_4574_nth__Cons__0,axiom,
    ! [X3: int,Xs: list_int] : aa_nat_int(nth_int(aa_list_int_list_int(cons_int(X3),Xs)),zero_zero_nat) = X3 ).

% nth_Cons_0
tff(fact_4575_length__nth__simps_I3_J,axiom,
    ! [X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi] : aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(X3,Xs)),zero_zero_nat) = X3 ).

% length_nth_simps(3)
tff(fact_4576_length__nth__simps_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] : aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs)),zero_zero_nat) = X3 ).

% length_nth_simps(3)
tff(fact_4577_length__nth__simps_I3_J,axiom,
    ! [X3: nat,Xs: list_nat] : aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(X3),Xs)),zero_zero_nat) = X3 ).

% length_nth_simps(3)
tff(fact_4578_length__nth__simps_I3_J,axiom,
    ! [X3: int,Xs: list_int] : aa_nat_int(nth_int(aa_list_int_list_int(cons_int(X3),Xs)),zero_zero_nat) = X3 ).

% length_nth_simps(3)
tff(fact_4579_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_4580_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% of_nat_mult
tff(fact_4581_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_4582_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_4583_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_4584_foldl__length,axiom,
    ! [L: list_real] : foldl_nat_real(aTP_Lamm_qn,zero_zero_nat,L) = aa_list_real_nat(size_size_list_real,L) ).

% foldl_length
tff(fact_4585_foldl__length,axiom,
    ! [L: list_bool] : foldl_nat_bool(aTP_Lamm_qo,zero_zero_nat,L) = aa_list_bool_nat(size_size_list_bool,L) ).

% foldl_length
tff(fact_4586_foldl__length,axiom,
    ! [L: list_nat] : aa_list_nat_nat(foldl_nat_nat(aTP_Lamm_qp,zero_zero_nat),L) = aa_list_nat_nat(size_size_list_nat,L) ).

% foldl_length
tff(fact_4587_foldl__length,axiom,
    ! [L: list_int] : foldl_nat_int(aTP_Lamm_qq,zero_zero_nat,L) = aa_list_int_nat(size_size_list_int,L) ).

% foldl_length
tff(fact_4588_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_4589_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_4590_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_4591_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_4592_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_4593_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_4594_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_4595_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_4596_horner__sum__simps_I2_J,axiom,
    ! [F: fun_nat_real,A: real,X3: nat,Xs: list_nat] : groups3482786445295563865t_real(F,A,aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),groups3482786445295563865t_real(F,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_4597_horner__sum__simps_I2_J,axiom,
    ! [F: fun_int_real,A: real,X3: int,Xs: list_int] : groups5669708019988585653t_real(F,A,aa_list_int_list_int(cons_int(X3),Xs)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(F,X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),groups5669708019988585653t_real(F,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_4598_horner__sum__simps_I2_J,axiom,
    ! [F: fun_nat_rat,A: rat,X3: nat,Xs: list_nat] : groups6853238114764508677at_rat(F,A,aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(F,X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),groups6853238114764508677at_rat(F,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_4599_horner__sum__simps_I2_J,axiom,
    ! [F: fun_int_rat,A: rat,X3: int,Xs: list_int] : groups7852591826665563233nt_rat(F,A,aa_list_int_list_int(cons_int(X3),Xs)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(F,X3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),groups7852591826665563233nt_rat(F,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_4600_horner__sum__simps_I2_J,axiom,
    ! [F: fun_nat_nat,A: nat,X3: nat,Xs: list_nat] : aa_list_nat_nat(groups7488368174851004413at_nat(F,A),aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_list_nat_nat(groups7488368174851004413at_nat(F,A),Xs))) ).

% horner_sum_simps(2)
tff(fact_4601_horner__sum__simps_I2_J,axiom,
    ! [F: fun_int_nat,A: nat,X3: int,Xs: list_int] : groups8487721886752058969nt_nat(F,A,aa_list_int_list_int(cons_int(X3),Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(F,X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),groups8487721886752058969nt_nat(F,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_4602_horner__sum__simps_I2_J,axiom,
    ! [F: fun_nat_int,A: int,X3: nat,Xs: list_nat] : groups7485877704341954137at_int(F,A,aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(F,X3)),aa_int_int(aa_int_fun_int_int(times_times_int,A),groups7485877704341954137at_int(F,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_4603_horner__sum__simps_I2_J,axiom,
    ! [F: fun_int_int,A: int,X3: int,Xs: list_int] : groups8485231416243008693nt_int(F,A,aa_list_int_list_int(cons_int(X3),Xs)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(F,X3)),aa_int_int(aa_int_fun_int_int(times_times_int,A),groups8485231416243008693nt_int(F,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_4604_horner__sum__simps_I2_J,axiom,
    ! [F: fun_nat_complex,A: complex,X3: nat,Xs: list_nat] : groups404637655443745499omplex(F,A,aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(F,X3)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),groups404637655443745499omplex(F,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_4605_horner__sum__simps_I2_J,axiom,
    ! [F: fun_int_complex,A: complex,X3: int,Xs: list_int] : groups1380173120649922871omplex(F,A,aa_list_int_list_int(cons_int(X3),Xs)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_int_complex(F,X3)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),groups1380173120649922871omplex(F,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_4606_enumerate__simps_I2_J,axiom,
    ! [N: nat,X3: int,Xs: list_int] : enumerate_int(N,aa_list_int_list_int(cons_int(X3),Xs)) = cons_P2335045147070616083at_int(product_Pair_nat_int(N,X3),enumerate_int(aa_nat_nat(suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_4607_enumerate__simps_I2_J,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] : enumerate_nat(N,aa_list_nat_list_nat(cons_nat(X3),Xs)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),X3),enumerate_nat(aa_nat_nat(suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_4608_enumerate__simps_I2_J,axiom,
    ! [N: nat,X3: num,Xs: list_num] : enumerate_num(N,cons_num(X3,Xs)) = cons_P7221140103528650497at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(N),X3),enumerate_num(aa_nat_nat(suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_4609_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_4610_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_4611_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_4612_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_4613_nth__Cons__pos,axiom,
    ! [N: nat,X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(X3,Xs)),N) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4614_nth__Cons__pos,axiom,
    ! [N: nat,X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs)),N) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4615_nth__Cons__pos,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(X3),Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4616_nth__Cons__pos,axiom,
    ! [N: nat,X3: int,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(nth_int(aa_list_int_list_int(cons_int(X3),Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4617_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X5: real] : pp(aa_set_real_bool(member_real(X5),S3))
     => ( ? [Z4: real] :
          ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),S3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Z4)) )
       => ? [Y3: real] :
            ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Y3)) )
            & ! [Z4: real] :
                ( ! [X: real] :
                    ( pp(aa_set_real_bool(member_real(X),S3))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Z4)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z4)) ) ) ) ) ).

% complete_real
tff(fact_4618_less__eq__real__def,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
        | ( X3 = Y ) ) ) ).

% less_eq_real_def
tff(fact_4619_reals__Archimedean3,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ! [Y4: real] :
        ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X3))) ) ).

% reals_Archimedean3
tff(fact_4620_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] : product_lists_nat(cons_list_nat(Xs,Xss)) = concat_list_nat(map_na6205611841492582150st_nat(aTP_Lamm_qr(Xss),Xs)) ).

% product_lists.simps(2)
tff(fact_4621_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_int,Xss: list_list_int] : product_lists_int(cons_list_int(Xs,Xss)) = concat_list_int(map_in1039701548059952062st_int(aTP_Lamm_qs(Xss),Xs)) ).

% product_lists.simps(2)
tff(fact_4622_real__arch__simple,axiom,
    ! [X3: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% real_arch_simple
tff(fact_4623_real__arch__simple,axiom,
    ! [X3: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% real_arch_simple
tff(fact_4624_reals__Archimedean2,axiom,
    ! [X3: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% reals_Archimedean2
tff(fact_4625_reals__Archimedean2,axiom,
    ! [X3: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% reals_Archimedean2
tff(fact_4626_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,X3)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,X3)) ).

% mult_of_nat_commute
tff(fact_4627_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,X3)),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),aa_nat_complex(semiri8010041392384452111omplex,X3)) ).

% mult_of_nat_commute
tff(fact_4628_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X3)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X3)) ).

% mult_of_nat_commute
tff(fact_4629_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X3)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X3)) ).

% mult_of_nat_commute
tff(fact_4630_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X3)) ).

% mult_of_nat_commute
tff(fact_4631_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% nat_less_real_le
tff(fact_4632_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_4633_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% of_nat_0_le_iff
tff(fact_4634_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% of_nat_0_le_iff
tff(fact_4635_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% of_nat_0_le_iff
tff(fact_4636_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% of_nat_0_le_iff
tff(fact_4637_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)) ).

% of_nat_less_0_iff
tff(fact_4638_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_4639_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_4640_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_4641_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N)) != zero_zero_complex ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4642_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4643_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4644_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4645_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4646_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_4647_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_4648_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_4649_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_4650_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).

% less_imp_of_nat_less
tff(fact_4651_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).

% less_imp_of_nat_less
tff(fact_4652_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% less_imp_of_nat_less
tff(fact_4653_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% less_imp_of_nat_less
tff(fact_4654_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).

% of_nat_mono
tff(fact_4655_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).

% of_nat_mono
tff(fact_4656_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).

% of_nat_mono
tff(fact_4657_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% of_nat_mono
tff(fact_4658_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,A)),aa_nat_rat(semiri681578069525770553at_rat,B)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4659_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,A)),aa_nat_real(semiri5074537144036343181t_real,B)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4660_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4661_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,A)),aa_nat_nat(semiri1316708129612266289at_nat,B)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4662_real__archimedian__rdiv__eq__0,axiom,
    ! [X3: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),X3)),C)) )
         => ( X3 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_4663_length__nth__simps_I2_J,axiom,
    ! [X3: real,Xs: list_real] : aa_list_real_nat(size_size_list_real,cons_real(X3,Xs)) = aa_nat_nat(suc,aa_list_real_nat(size_size_list_real,Xs)) ).

% length_nth_simps(2)
tff(fact_4664_length__nth__simps_I2_J,axiom,
    ! [X3: bool,Xs: list_bool] : aa_list_bool_nat(size_size_list_bool,cons_bool(X3,Xs)) = aa_nat_nat(suc,aa_list_bool_nat(size_size_list_bool,Xs)) ).

% length_nth_simps(2)
tff(fact_4665_length__nth__simps_I2_J,axiom,
    ! [X3: nat,Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_nat_nat(suc,aa_list_nat_nat(size_size_list_nat,Xs)) ).

% length_nth_simps(2)
tff(fact_4666_length__nth__simps_I2_J,axiom,
    ! [X3: int,Xs: list_int] : aa_list_int_nat(size_size_list_int,aa_list_int_list_int(cons_int(X3),Xs)) = aa_nat_nat(suc,aa_list_int_nat(size_size_list_int,Xs)) ).

% length_nth_simps(2)
tff(fact_4667_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( aa_nat_nat(suc,N) = aa_list_real_nat(size_size_list_real,Xs) )
    <=> ? [Y5: real,Ys3: list_real] :
          ( ( Xs = cons_real(Y5,Ys3) )
          & ( aa_list_real_nat(size_size_list_real,Ys3) = N ) ) ) ).

% Suc_length_conv
tff(fact_4668_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( aa_nat_nat(suc,N) = aa_list_bool_nat(size_size_list_bool,Xs) )
    <=> ? [Y5: bool,Ys3: list_bool] :
          ( ( Xs = cons_bool(Y5,Ys3) )
          & ( aa_list_bool_nat(size_size_list_bool,Ys3) = N ) ) ) ).

% Suc_length_conv
tff(fact_4669_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( aa_nat_nat(suc,N) = aa_list_nat_nat(size_size_list_nat,Xs) )
    <=> ? [Y5: nat,Ys3: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(cons_nat(Y5),Ys3) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys3) = N ) ) ) ).

% Suc_length_conv
tff(fact_4670_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( aa_nat_nat(suc,N) = aa_list_int_nat(size_size_list_int,Xs) )
    <=> ? [Y5: int,Ys3: list_int] :
          ( ( Xs = aa_list_int_list_int(cons_int(Y5),Ys3) )
          & ( aa_list_int_nat(size_size_list_int,Ys3) = N ) ) ) ).

% Suc_length_conv
tff(fact_4671_length__Suc__conv,axiom,
    ! [Xs: list_real,N: nat] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: real,Ys3: list_real] :
          ( ( Xs = cons_real(Y5,Ys3) )
          & ( aa_list_real_nat(size_size_list_real,Ys3) = N ) ) ) ).

% length_Suc_conv
tff(fact_4672_length__Suc__conv,axiom,
    ! [Xs: list_bool,N: nat] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: bool,Ys3: list_bool] :
          ( ( Xs = cons_bool(Y5,Ys3) )
          & ( aa_list_bool_nat(size_size_list_bool,Ys3) = N ) ) ) ).

% length_Suc_conv
tff(fact_4673_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: nat,Ys3: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(cons_nat(Y5),Ys3) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys3) = N ) ) ) ).

% length_Suc_conv
tff(fact_4674_length__Suc__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: int,Ys3: list_int] :
          ( ( Xs = aa_list_int_list_int(cons_int(Y5),Ys3) )
          & ( aa_list_int_nat(size_size_list_int,Ys3) = N ) ) ) ).

% length_Suc_conv
tff(fact_4675_impossible__Cons,axiom,
    ! [Xs: list_real,Ys: list_real,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_real_nat(size_size_list_real,Ys)))
     => ( Xs != cons_real(X3,Ys) ) ) ).

% impossible_Cons
tff(fact_4676_impossible__Cons,axiom,
    ! [Xs: list_bool,Ys: list_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),aa_list_bool_nat(size_size_list_bool,Ys)))
     => ( Xs != cons_bool(X3,Ys) ) ) ).

% impossible_Cons
tff(fact_4677_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)))
     => ( Xs != aa_list_nat_list_nat(cons_nat(X3),Ys) ) ) ).

% impossible_Cons
tff(fact_4678_impossible__Cons,axiom,
    ! [Xs: list_int,Ys: list_int,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_int_nat(size_size_list_int,Ys)))
     => ( Xs != aa_list_int_list_int(cons_int(X3),Ys) ) ) ).

% impossible_Cons
tff(fact_4679_set__subset__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,cons_VEBT_VEBT(X3,Xs)))) ).

% set_subset_Cons
tff(fact_4680_set__subset__Cons,axiom,
    ! [Xs: list_real,X3: real] : pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),aa_lis3769085826154824284t_real(set_real2,cons_real(X3,Xs)))) ).

% set_subset_Cons
tff(fact_4681_set__subset__Cons,axiom,
    ! [Xs: list_bool,X3: bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs)),aa_lis3587183946281582166t_bool(set_bool2,cons_bool(X3,Xs)))) ).

% set_subset_Cons
tff(fact_4682_set__subset__Cons,axiom,
    ! [Xs: list_nat,X3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Xs)),aa_list_nat_set_nat(set_nat2,aa_list_nat_list_nat(cons_nat(X3),Xs)))) ).

% set_subset_Cons
tff(fact_4683_set__subset__Cons,axiom,
    ! [Xs: list_int,X3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Xs)),aa_list_int_set_int(set_int2,aa_list_int_list_int(cons_int(X3),Xs)))) ).

% set_subset_Cons
tff(fact_4684_sorted2,axiom,
    ! [X3: rat,Y: rat,Zs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,cons_rat(X3,cons_rat(Y,Zs)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
        & sorted_wrt_rat(ord_less_eq_rat,cons_rat(Y,Zs)) ) ) ).

% sorted2
tff(fact_4685_sorted2,axiom,
    ! [X3: num,Y: num,Zs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,cons_num(X3,cons_num(Y,Zs)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
        & sorted_wrt_num(ord_less_eq_num,cons_num(Y,Zs)) ) ) ).

% sorted2
tff(fact_4686_sorted2,axiom,
    ! [X3: nat,Y: nat,Zs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),aa_list_nat_list_nat(cons_nat(X3),aa_list_nat_list_nat(cons_nat(Y),Zs))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
        & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),aa_list_nat_list_nat(cons_nat(Y),Zs))) ) ) ).

% sorted2
tff(fact_4687_sorted2,axiom,
    ! [X3: int,Y: int,Zs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),aa_list_int_list_int(cons_int(X3),aa_list_int_list_int(cons_int(Y),Zs))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
        & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),aa_list_int_list_int(cons_int(Y),Zs))) ) ) ).

% sorted2
tff(fact_4688_list__update__code_I3_J,axiom,
    ! [X3: nat,Xs: list_nat,I: nat,Y: nat] : list_update_nat(aa_list_nat_list_nat(cons_nat(X3),Xs),aa_nat_nat(suc,I),Y) = aa_list_nat_list_nat(cons_nat(X3),list_update_nat(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_4689_list__update__code_I3_J,axiom,
    ! [X3: int,Xs: list_int,I: nat,Y: int] : list_update_int(aa_list_int_list_int(cons_int(X3),Xs),aa_nat_nat(suc,I),Y) = aa_list_int_list_int(cons_int(X3),list_update_int(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_4690_list__update__code_I3_J,axiom,
    ! [X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] : list_u6098035379799741383_VEBTi(cons_VEBT_VEBTi(X3,Xs),aa_nat_nat(suc,I),Y) = cons_VEBT_VEBTi(X3,list_u6098035379799741383_VEBTi(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_4691_list__update__code_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] : aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(X3,Xs),aa_nat_nat(suc,I)),Y) = cons_VEBT_VEBT(X3,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),Y)) ).

% list_update_code(3)
tff(fact_4692_list__update__code_I2_J,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat] : list_update_nat(aa_list_nat_list_nat(cons_nat(X3),Xs),zero_zero_nat,Y) = aa_list_nat_list_nat(cons_nat(Y),Xs) ).

% list_update_code(2)
tff(fact_4693_list__update__code_I2_J,axiom,
    ! [X3: int,Xs: list_int,Y: int] : list_update_int(aa_list_int_list_int(cons_int(X3),Xs),zero_zero_nat,Y) = aa_list_int_list_int(cons_int(Y),Xs) ).

% list_update_code(2)
tff(fact_4694_list__update__code_I2_J,axiom,
    ! [X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi,Y: vEBT_VEBTi] : list_u6098035379799741383_VEBTi(cons_VEBT_VEBTi(X3,Xs),zero_zero_nat,Y) = cons_VEBT_VEBTi(Y,Xs) ).

% list_update_code(2)
tff(fact_4695_list__update__code_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] : aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(X3,Xs),zero_zero_nat),Y) = cons_VEBT_VEBT(Y,Xs) ).

% list_update_code(2)
tff(fact_4696_replicate__Suc,axiom,
    ! [N: nat,X3: nat] : replicate_nat(aa_nat_nat(suc,N),X3) = aa_list_nat_list_nat(cons_nat(X3),replicate_nat(N,X3)) ).

% replicate_Suc
tff(fact_4697_replicate__Suc,axiom,
    ! [N: nat,X3: int] : replicate_int(aa_nat_nat(suc,N),X3) = aa_list_int_list_int(cons_int(X3),replicate_int(N,X3)) ).

% replicate_Suc
tff(fact_4698_replicate__Suc,axiom,
    ! [N: nat,X3: vEBT_VEBT] : aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(suc,N)),X3) = cons_VEBT_VEBT(X3,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3)) ).

% replicate_Suc
tff(fact_4699_replicate__Suc,axiom,
    ! [N: nat,X3: bool] : replicate_bool(aa_nat_nat(suc,N),X3) = cons_bool(X3,replicate_bool(N,X3)) ).

% replicate_Suc
tff(fact_4700_foldl__absorb1,axiom,
    ! [X3: real,Zs: list_real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),foldl_real_real(times_times_real,one_one_real,Zs)) = foldl_real_real(times_times_real,X3,Zs) ).

% foldl_absorb1
tff(fact_4701_foldl__absorb1,axiom,
    ! [X3: rat,Zs: list_rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),foldl_rat_rat(times_times_rat,one_one_rat,Zs)) = foldl_rat_rat(times_times_rat,X3,Zs) ).

% foldl_absorb1
tff(fact_4702_foldl__absorb1,axiom,
    ! [X3: nat,Zs: list_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),aa_list_nat_nat(foldl_nat_nat(times_times_nat,one_one_nat),Zs)) = aa_list_nat_nat(foldl_nat_nat(times_times_nat,X3),Zs) ).

% foldl_absorb1
tff(fact_4703_foldl__absorb1,axiom,
    ! [X3: int,Zs: list_int] : aa_int_int(aa_int_fun_int_int(times_times_int,X3),foldl_int_int(times_times_int,one_one_int,Zs)) = foldl_int_int(times_times_int,X3,Zs) ).

% foldl_absorb1
tff(fact_4704_foldl__absorb1,axiom,
    ! [X3: assn,Zs: list_assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X3),foldl_assn_assn(times_times_assn,one_one_assn,Zs)) = foldl_assn_assn(times_times_assn,X3,Zs) ).

% foldl_absorb1
tff(fact_4705_foldl__absorb1,axiom,
    ! [X3: complex,Zs: list_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),foldl_2274445284955862271omplex(times_times_complex,one_one_complex,Zs)) = foldl_2274445284955862271omplex(times_times_complex,X3,Zs) ).

% foldl_absorb1
tff(fact_4706_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_nat_fun_nat_assn,A: nat,As: list_nat,C: nat,Cs: list_nat] : vEBT_L8301102511889123557at_nat(P,aa_list_nat_list_nat(cons_nat(A),As),aa_list_nat_list_nat(cons_nat(C),Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(P,A),C)),vEBT_L8301102511889123557at_nat(P,As,Cs)) ).

% list_assn.simps(2)
tff(fact_4707_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_nat_fun_int_assn,A: nat,As: list_nat,C: int,Cs: list_int] : vEBT_L8298612041380073281at_int(P,aa_list_nat_list_nat(cons_nat(A),As),aa_list_int_list_int(cons_int(C),Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(P,A),C)),vEBT_L8298612041380073281at_int(P,As,Cs)) ).

% list_assn.simps(2)
tff(fact_4708_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_int_fun_nat_assn,A: int,As: list_int,C: nat,Cs: list_nat] : vEBT_L77084186935402305nt_nat(P,aa_list_int_list_int(cons_int(A),As),aa_list_nat_list_nat(cons_nat(C),Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(P,A),C)),vEBT_L77084186935402305nt_nat(P,As,Cs)) ).

% list_assn.simps(2)
tff(fact_4709_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_int_fun_int_assn,A: int,As: list_int,C: int,Cs: list_int] : vEBT_L74593716426352029nt_int(P,aa_list_int_list_int(cons_int(A),As),aa_list_int_list_int(cons_int(C),Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(P,A),C)),vEBT_L74593716426352029nt_int(P,As,Cs)) ).

% list_assn.simps(2)
tff(fact_4710_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_VE905575822219544865i_assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P,cons_VEBT_VEBT(A,As)),cons_VEBT_VEBTi(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P,A),C)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P,As),Cs)) ).

% list_assn.simps(2)
tff(fact_4711_list__assn__simps_I2_J,axiom,
    ! [P: fun_nat_fun_nat_assn,A: nat,As: list_nat,C: nat,Cs: list_nat] : vEBT_L8301102511889123557at_nat(P,aa_list_nat_list_nat(cons_nat(A),As),aa_list_nat_list_nat(cons_nat(C),Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(P,A),C)),vEBT_L8301102511889123557at_nat(P,As,Cs)) ).

% list_assn_simps(2)
tff(fact_4712_list__assn__simps_I2_J,axiom,
    ! [P: fun_nat_fun_int_assn,A: nat,As: list_nat,C: int,Cs: list_int] : vEBT_L8298612041380073281at_int(P,aa_list_nat_list_nat(cons_nat(A),As),aa_list_int_list_int(cons_int(C),Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(P,A),C)),vEBT_L8298612041380073281at_int(P,As,Cs)) ).

% list_assn_simps(2)
tff(fact_4713_list__assn__simps_I2_J,axiom,
    ! [P: fun_int_fun_nat_assn,A: int,As: list_int,C: nat,Cs: list_nat] : vEBT_L77084186935402305nt_nat(P,aa_list_int_list_int(cons_int(A),As),aa_list_nat_list_nat(cons_nat(C),Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(P,A),C)),vEBT_L77084186935402305nt_nat(P,As,Cs)) ).

% list_assn_simps(2)
tff(fact_4714_list__assn__simps_I2_J,axiom,
    ! [P: fun_int_fun_int_assn,A: int,As: list_int,C: int,Cs: list_int] : vEBT_L74593716426352029nt_int(P,aa_list_int_list_int(cons_int(A),As),aa_list_int_list_int(cons_int(C),Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(P,A),C)),vEBT_L74593716426352029nt_int(P,As,Cs)) ).

% list_assn_simps(2)
tff(fact_4715_list__assn__simps_I2_J,axiom,
    ! [P: fun_VE905575822219544865i_assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P,cons_VEBT_VEBT(A,As)),cons_VEBT_VEBTi(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P,A),C)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P,As),Cs)) ).

% list_assn_simps(2)
tff(fact_4716_ex__less__of__nat__mult,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X3))
     => ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),X3))) ) ).

% ex_less_of_nat_mult
tff(fact_4717_ex__less__of__nat__mult,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X3))) ) ).

% ex_less_of_nat_mult
tff(fact_4718_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,K) != zero_zero_complex )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4719_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,K) != zero_zero_rat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4720_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,K) != zero_zero_real )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4721_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,K) != zero_zero_int )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4722_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,K) != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4723_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_4724_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_4725_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_4726_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_4727_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_list_real_nat(size_size_list_real,Xs)))
    <=> ? [X4: real,Ys3: list_real] :
          ( ( Xs = cons_real(X4,Ys3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_real_nat(size_size_list_real,Ys3))) ) ) ).

% Suc_le_length_iff
tff(fact_4728_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_list_bool_nat(size_size_list_bool,Xs)))
    <=> ? [X4: bool,Ys3: list_bool] :
          ( ( Xs = cons_bool(X4,Ys3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_bool_nat(size_size_list_bool,Ys3))) ) ) ).

% Suc_le_length_iff
tff(fact_4729_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_list_nat_nat(size_size_list_nat,Xs)))
    <=> ? [X4: nat,Ys3: list_nat] :
          ( ( Xs = aa_list_nat_list_nat(cons_nat(X4),Ys3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_nat_nat(size_size_list_nat,Ys3))) ) ) ).

% Suc_le_length_iff
tff(fact_4730_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_list_int_nat(size_size_list_int,Xs)))
    <=> ? [X4: int,Ys3: list_int] :
          ( ( Xs = aa_list_int_list_int(cons_int(X4),Ys3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_int_nat(size_size_list_int,Ys3))) ) ) ).

% Suc_le_length_iff
tff(fact_4731_sorted__simps_I2_J,axiom,
    ! [X3: real,Ys: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),cons_real(X3,Ys)))
    <=> ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),aa_lis3769085826154824284t_real(set_real2,Ys)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),X4)) )
        & pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Ys)) ) ) ).

% sorted_simps(2)
tff(fact_4732_sorted__simps_I2_J,axiom,
    ! [X3: bool,Ys: list_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),cons_bool(X3,Ys)))
    <=> ( ! [X4: bool] :
            ( pp(aa_set_bool_bool(member_bool(X4),aa_lis3587183946281582166t_bool(set_bool2,Ys)))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X3),X4)) )
        & pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Ys)) ) ) ).

% sorted_simps(2)
tff(fact_4733_sorted__simps_I2_J,axiom,
    ! [X3: rat,Ys: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,cons_rat(X3,Ys))
    <=> ( ! [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),set_rat2(Ys)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),X4)) )
        & sorted_wrt_rat(ord_less_eq_rat,Ys) ) ) ).

% sorted_simps(2)
tff(fact_4734_sorted__simps_I2_J,axiom,
    ! [X3: num,Ys: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,cons_num(X3,Ys))
    <=> ( ! [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),set_num2(Ys)))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),X4)) )
        & sorted_wrt_num(ord_less_eq_num,Ys) ) ) ).

% sorted_simps(2)
tff(fact_4735_sorted__simps_I2_J,axiom,
    ! [X3: nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),aa_list_nat_list_nat(cons_nat(X3),Ys)))
    <=> ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),aa_list_nat_set_nat(set_nat2,Ys)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),X4)) )
        & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Ys)) ) ) ).

% sorted_simps(2)
tff(fact_4736_sorted__simps_I2_J,axiom,
    ! [X3: int,Ys: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),aa_list_int_list_int(cons_int(X3),Ys)))
    <=> ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),aa_list_int_set_int(set_int2,Ys)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),X4)) )
        & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Ys)) ) ) ).

% sorted_simps(2)
tff(fact_4737_strict__sorted__simps_I2_J,axiom,
    ! [X3: bool,Ys: list_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_bool),cons_bool(X3,Ys)))
    <=> ( ! [X4: bool] :
            ( pp(aa_set_bool_bool(member_bool(X4),aa_lis3587183946281582166t_bool(set_bool2,Ys)))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X3),X4)) )
        & pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_bool),Ys)) ) ) ).

% strict_sorted_simps(2)
tff(fact_4738_strict__sorted__simps_I2_J,axiom,
    ! [X3: real,Ys: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_real),cons_real(X3,Ys)))
    <=> ( ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),aa_lis3769085826154824284t_real(set_real2,Ys)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),X4)) )
        & pp(aa_list_real_bool(sorted_wrt_real(ord_less_real),Ys)) ) ) ).

% strict_sorted_simps(2)
tff(fact_4739_strict__sorted__simps_I2_J,axiom,
    ! [X3: rat,Ys: list_rat] :
      ( sorted_wrt_rat(ord_less_rat,cons_rat(X3,Ys))
    <=> ( ! [X4: rat] :
            ( pp(aa_set_rat_bool(member_rat(X4),set_rat2(Ys)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),X4)) )
        & sorted_wrt_rat(ord_less_rat,Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_4740_strict__sorted__simps_I2_J,axiom,
    ! [X3: num,Ys: list_num] :
      ( sorted_wrt_num(ord_less_num,cons_num(X3,Ys))
    <=> ( ! [X4: num] :
            ( pp(aa_set_num_bool(member_num(X4),set_num2(Ys)))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),X4)) )
        & sorted_wrt_num(ord_less_num,Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_4741_strict__sorted__simps_I2_J,axiom,
    ! [X3: nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),aa_list_nat_list_nat(cons_nat(X3),Ys)))
    <=> ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),aa_list_nat_set_nat(set_nat2,Ys)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X4)) )
        & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ys)) ) ) ).

% strict_sorted_simps(2)
tff(fact_4742_strict__sorted__simps_I2_J,axiom,
    ! [X3: int,Ys: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),aa_list_int_list_int(cons_int(X3),Ys)))
    <=> ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),aa_list_int_set_int(set_int2,Ys)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),X4)) )
        & pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),Ys)) ) ) ).

% strict_sorted_simps(2)
tff(fact_4743_TBOUND__of__list,axiom,
    ! [Xs: list_bool] : pp(aa_nat_bool(time_T4182777596009358560y_bool(array_of_list_bool(Xs)),aa_nat_nat(suc,aa_list_bool_nat(size_size_list_bool,Xs)))) ).

% TBOUND_of_list
tff(fact_4744_TBOUND__of__list,axiom,
    ! [Xs: list_nat] : pp(aa_nat_bool(time_T3621710982123377501ay_nat(array_of_list_nat(Xs)),aa_nat_nat(suc,aa_list_nat_nat(size_size_list_nat,Xs)))) ).

% TBOUND_of_list
tff(fact_4745_TBOUND__of__list,axiom,
    ! [Xs: list_int] : pp(aa_nat_bool(time_T8667231999468956601ay_int(array_of_list_int(Xs)),aa_nat_nat(suc,aa_list_int_nat(size_size_list_int,Xs)))) ).

% TBOUND_of_list
tff(fact_4746_foldl__length__aux,axiom,
    ! [A: nat,L: list_real] : foldl_nat_real(aTP_Lamm_qn,A,L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_real_nat(size_size_list_real,L)) ).

% foldl_length_aux
tff(fact_4747_foldl__length__aux,axiom,
    ! [A: nat,L: list_bool] : foldl_nat_bool(aTP_Lamm_qo,A,L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_bool_nat(size_size_list_bool,L)) ).

% foldl_length_aux
tff(fact_4748_foldl__length__aux,axiom,
    ! [A: nat,L: list_nat] : aa_list_nat_nat(foldl_nat_nat(aTP_Lamm_qp,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_4749_foldl__length__aux,axiom,
    ! [A: nat,L: list_int] : foldl_nat_int(aTP_Lamm_qq,A,L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_int_nat(size_size_list_int,L)) ).

% foldl_length_aux
tff(fact_4750_list_Osize_I4_J,axiom,
    ! [X21: real,X22: list_real] : aa_list_real_nat(size_size_list_real,cons_real(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_real_nat(size_size_list_real,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_4751_list_Osize_I4_J,axiom,
    ! [X21: bool,X22: list_bool] : aa_list_bool_nat(size_size_list_bool,cons_bool(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_bool_nat(size_size_list_bool,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_4752_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] : aa_list_nat_nat(size_size_list_nat,aa_list_nat_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_4753_list_Osize_I4_J,axiom,
    ! [X21: int,X22: list_int] : aa_list_int_nat(size_size_list_int,aa_list_int_list_int(cons_int(X21),X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_int_nat(size_size_list_int,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_4754_nth__Cons_H,axiom,
    ! [N: nat,X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(X3,Xs)),N) = X3 ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(X3,Xs)),N) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_4755_nth__Cons_H,axiom,
    ! [N: nat,X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs)),N) = X3 ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs)),N) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_4756_nth__Cons_H,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(X3),Xs)),N) = X3 ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(X3),Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_4757_nth__Cons_H,axiom,
    ! [N: nat,X3: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(nth_int(aa_list_int_list_int(cons_int(X3),Xs)),N) = X3 ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(nth_int(aa_list_int_list_int(cons_int(X3),Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_4758_n__lists_Osimps_I2_J,axiom,
    ! [N: nat,Xs: list_nat] : n_lists_nat(aa_nat_nat(suc,N),Xs) = concat_list_nat(map_li960784813134754710st_nat(aTP_Lamm_qu(Xs),n_lists_nat(N,Xs))) ).

% n_lists.simps(2)
tff(fact_4759_n__lists_Osimps_I2_J,axiom,
    ! [N: nat,Xs: list_int] : n_lists_int(aa_nat_nat(suc,N),Xs) = concat_list_int(map_li8902190837986183758st_int(aTP_Lamm_qw(Xs),n_lists_int(N,Xs))) ).

% n_lists.simps(2)
tff(fact_4760_Id__on__fold,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( id_on_complex(A2) = finite908486625158157349omplex(aTP_Lamm_qx,bot_bo313579480278226313omplex,A2) ) ) ).

% Id_on_fold
tff(fact_4761_Id__on__fold,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( id_on_nat(A2) = finite3745491028973389255at_nat(aTP_Lamm_qy,bot_bo2099793752762293965at_nat,A2) ) ) ).

% Id_on_fold
tff(fact_4762_Id__on__fold,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( id_on_int(A2) = finite5202366122487795491nt_int(aTP_Lamm_qz,bot_bo1796632182523588997nt_int,A2) ) ) ).

% Id_on_fold
tff(fact_4763_Id__on__fold,axiom,
    ! [A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( id_on_Code_integer(A2) = finite6473783728727339668nteger(aTP_Lamm_ra,bot_bo4276436098303576167nteger,A2) ) ) ).

% Id_on_fold
tff(fact_4764_list_Osize__gen_I2_J,axiom,
    ! [X3: fun_nat_nat,X21: nat,X22: list_nat] : aa_list_nat_nat(size_list_nat(X3),aa_list_nat_list_nat(cons_nat(X21),X22)) = 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(X3,X21)),aa_list_nat_nat(size_list_nat(X3),X22))),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size_gen(2)
tff(fact_4765_list_Osize__gen_I2_J,axiom,
    ! [X3: fun_int_nat,X21: int,X22: list_int] : size_list_int(X3,aa_list_int_list_int(cons_int(X21),X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(X3,X21)),size_list_int(X3,X22))),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size_gen(2)
tff(fact_4766_list_Osize__gen_I2_J,axiom,
    ! [X3: fun_VEBT_VEBT_nat,X21: vEBT_VEBT,X22: list_VEBT_VEBT] : size_list_VEBT_VEBT(X3,cons_VEBT_VEBT(X21,X22)) = 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(X3,X21)),size_list_VEBT_VEBT(X3,X22))),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size_gen(2)
tff(fact_4767_nth__non__equal__first__eq,axiom,
    ! [X3: vEBT_VEBTi,Y: vEBT_VEBTi,Xs: list_VEBT_VEBTi,N: nat] :
      ( ( X3 != Y )
     => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(X3,Xs)),N) = Y )
      <=> ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4768_nth__non__equal__first__eq,axiom,
    ! [X3: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ( X3 != Y )
     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs)),N) = Y )
      <=> ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4769_nth__non__equal__first__eq,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X3 != Y )
     => ( ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(X3),Xs)),N) = Y )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4770_nth__non__equal__first__eq,axiom,
    ! [X3: int,Y: int,Xs: list_int,N: nat] :
      ( ( X3 != Y )
     => ( ( aa_nat_int(nth_int(aa_list_int_list_int(cons_int(X3),Xs)),N) = Y )
      <=> ( ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4771_nth__equal__first__eq,axiom,
    ! [X3: vEBT_VEBTi,Xs: list_VEBT_VEBTi,N: nat] :
      ( ~ member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s7982070591426661849_VEBTi(Xs)))
       => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(X3,Xs)),N) = X3 )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4772_nth__equal__first__eq,axiom,
    ! [X3: set_nat,Xs: list_set_nat,N: nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s3254054031482475050et_nat(Xs)))
       => ( ( aa_nat_set_nat(nth_set_nat(cons_set_nat(X3,Xs)),N) = X3 )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4773_nth__equal__first__eq,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(X3,Xs)),N) = X3 )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4774_nth__equal__first__eq,axiom,
    ! [X3: real,Xs: list_real,N: nat] :
      ( ~ pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
       => ( ( aa_nat_real(nth_real(cons_real(X3,Xs)),N) = X3 )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4775_nth__equal__first__eq,axiom,
    ! [X3: bool,Xs: list_bool,N: nat] :
      ( ~ pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( ( pp(aa_nat_bool(nth_bool(cons_bool(X3,Xs)),N))
          <=> pp(X3) )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4776_nth__equal__first__eq,axiom,
    ! [X3: nat,Xs: list_nat,N: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(X3),Xs)),N) = X3 )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4777_nth__equal__first__eq,axiom,
    ! [X3: int,Xs: list_int,N: nat] :
      ( ~ pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
       => ( ( aa_nat_int(nth_int(aa_list_int_list_int(cons_int(X3),Xs)),N) = X3 )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4778_Cons__replicate__eq,axiom,
    ! [X3: nat,Xs: list_nat,N: nat,Y: nat] :
      ( ( aa_list_nat_list_nat(cons_nat(X3),Xs) = replicate_nat(N,Y) )
    <=> ( ( X3 = Y )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( Xs = replicate_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),X3) ) ) ) ).

% Cons_replicate_eq
tff(fact_4779_Cons__replicate__eq,axiom,
    ! [X3: int,Xs: list_int,N: nat,Y: int] :
      ( ( aa_list_int_list_int(cons_int(X3),Xs) = replicate_int(N,Y) )
    <=> ( ( X3 = Y )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( Xs = replicate_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),X3) ) ) ) ).

% Cons_replicate_eq
tff(fact_4780_Cons__replicate__eq,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( cons_VEBT_VEBT(X3,Xs) = aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),Y) )
    <=> ( ( X3 = Y )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( Xs = aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),X3) ) ) ) ).

% Cons_replicate_eq
tff(fact_4781_Cons__replicate__eq,axiom,
    ! [X3: bool,Xs: list_bool,N: nat,Y: bool] :
      ( ( cons_bool(X3,Xs) = replicate_bool(N,Y) )
    <=> ( ( pp(X3)
        <=> pp(Y) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( Xs = replicate_bool(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),X3) ) ) ) ).

% Cons_replicate_eq
tff(fact_4782_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X3: nat,Xs: list_nat] :
      ( ( ( ( Begin = zero_zero_nat )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_nat(Begin,End,aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_list_nat_list_nat(cons_nat(X3),slice_nat(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_nat(Begin,End,aa_list_nat_list_nat(cons_nat(X3),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_4783_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X3: int,Xs: list_int] :
      ( ( ( ( Begin = zero_zero_nat )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_int(Begin,End,aa_list_int_list_int(cons_int(X3),Xs)) = aa_list_int_list_int(cons_int(X3),slice_int(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_int(Begin,End,aa_list_int_list_int(cons_int(X3),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_4784_time__array__of__list,axiom,
    ! [Xs: list_bool,H: heap_e7401611519738050253t_unit] : time_time_array_bool(array_of_list_bool(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_bool_nat(size_size_list_bool,Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_4785_time__array__of__list,axiom,
    ! [Xs: list_nat,H: heap_e7401611519738050253t_unit] : time_time_array_nat(array_of_list_nat(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_4786_time__array__of__list,axiom,
    ! [Xs: list_int,H: heap_e7401611519738050253t_unit] : time_time_array_int(array_of_list_int(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_int_nat(size_size_list_int,Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_4787_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_4788_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T))) ).

% cnt_non_neg
tff(fact_4789_not__real__square__gt__zero,axiom,
    ! [X3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3)))
    <=> ( X3 = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_4790_of__nat__code,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,N) = aa_complex_complex(aa_nat3829984526574172007omplex(semiri2816024913162550771omplex(aTP_Lamm_rb),N),zero_zero_complex) ).

% of_nat_code
tff(fact_4791_of__nat__code,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_rc,N),zero_zero_rat) ).

% of_nat_code
tff(fact_4792_of__nat__code,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(semiri7260567687927622513x_real(aTP_Lamm_rd,N),zero_zero_real) ).

% of_nat_code
tff(fact_4793_of__nat__code,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_re,N),zero_zero_int) ).

% of_nat_code
tff(fact_4794_of__nat__code,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_rf),N),zero_zero_nat) ).

% of_nat_code
tff(fact_4795_length__Cons,axiom,
    ! [X3: real,Xs: list_real] : aa_list_real_nat(size_size_list_real,cons_real(X3,Xs)) = aa_nat_nat(suc,aa_list_real_nat(size_size_list_real,Xs)) ).

% length_Cons
tff(fact_4796_length__Cons,axiom,
    ! [X3: bool,Xs: list_bool] : aa_list_bool_nat(size_size_list_bool,cons_bool(X3,Xs)) = aa_nat_nat(suc,aa_list_bool_nat(size_size_list_bool,Xs)) ).

% length_Cons
tff(fact_4797_length__Cons,axiom,
    ! [X3: nat,Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_nat_nat(suc,aa_list_nat_nat(size_size_list_nat,Xs)) ).

% length_Cons
tff(fact_4798_length__Cons,axiom,
    ! [X3: int,Xs: list_int] : aa_list_int_nat(size_size_list_int,aa_list_int_list_int(cons_int(X3),Xs)) = aa_nat_nat(suc,aa_list_int_nat(size_size_list_int,Xs)) ).

% length_Cons
tff(fact_4799_pochhammer__code,axiom,
    ! [N: nat,A: rat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_rg(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_rat) ) ) ) ).

% pochhammer_code
tff(fact_4800_pochhammer__code,axiom,
    ! [N: nat,A: complex] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_rh(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_complex) ) ) ) ).

% pochhammer_code
tff(fact_4801_pochhammer__code,axiom,
    ! [N: nat,A: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = aa_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ri(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_real) ) ) ) ).

% pochhammer_code
tff(fact_4802_pochhammer__code,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_rj(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_int) ) ) ) ).

% pochhammer_code
tff(fact_4803_pochhammer__code,axiom,
    ! [N: nat,A: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_rk(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat) ) ) ) ).

% pochhammer_code
tff(fact_4804_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,X3) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3 = 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_4805_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(comm_s7457072308508201937r_real(A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_4806_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_4807_pochhammer__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_4808_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_4809_pochhammer__of__nat,axiom,
    ! [X3: nat,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_nat_real(semiri5074537144036343181t_real,X3)),N) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X3),N)) ).

% pochhammer_of_nat
tff(fact_4810_pochhammer__of__nat,axiom,
    ! [X3: nat,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_nat_int(semiri1314217659103216013at_int,X3)),N) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X3),N)) ).

% pochhammer_of_nat
tff(fact_4811_pochhammer__of__nat,axiom,
    ! [X3: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),N) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X3),N)) ).

% pochhammer_of_nat
tff(fact_4812_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
    <=> ? [N6: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N6))) ) ).

% zless_iff_Suc_zadd
tff(fact_4813_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_4814_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_4815_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_4816_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(2)
tff(fact_4817_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(3)
tff(fact_4818_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% zle_int
tff(fact_4819_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = 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_4820_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_4821_nat__less__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ) ).

% nat_less_as_int
tff(fact_4822_nat__leq__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ) ).

% nat_leq_as_int
tff(fact_4823_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = one_one_real ).

% VEBT_internal.cnt.simps(1)
tff(fact_4824_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% pos_int_cases
tff(fact_4825_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_4826_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_4827_pochhammer__pos,axiom,
    ! [X3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X3),N))) ) ).

% pochhammer_pos
tff(fact_4828_pochhammer__pos,axiom,
    ! [X3: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X3),N))) ) ).

% pochhammer_pos
tff(fact_4829_pochhammer__pos,axiom,
    ! [X3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X3),N))) ) ).

% pochhammer_pos
tff(fact_4830_pochhammer__pos,axiom,
    ! [X3: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X3),N))) ) ).

% pochhammer_pos
tff(fact_4831_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_4832_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_4833_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_4834_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_4835_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),M) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_4836_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_4837_pochhammer__nonneg,axiom,
    ! [X3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X3),N))) ) ).

% pochhammer_nonneg
tff(fact_4838_pochhammer__nonneg,axiom,
    ! [X3: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X3),N))) ) ).

% pochhammer_nonneg
tff(fact_4839_pochhammer__nonneg,axiom,
    ! [X3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X3),N))) ) ).

% pochhammer_nonneg
tff(fact_4840_pochhammer__nonneg,axiom,
    ! [X3: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X3),N))) ) ).

% pochhammer_nonneg
tff(fact_4841_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X3: nat,Y: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_4842_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% pochhammer_0_left
tff(fact_4843_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N) = zero_zero_real ) ) ) ).

% pochhammer_0_left
tff(fact_4844_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% pochhammer_0_left
tff(fact_4845_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% pochhammer_0_left
tff(fact_4846_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% pochhammer_0_left
tff(fact_4847_pochhammer__rec,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),N)) ).

% pochhammer_rec
tff(fact_4848_pochhammer__rec,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),N)) ).

% pochhammer_rec
tff(fact_4849_pochhammer__rec,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),N)) ).

% pochhammer_rec
tff(fact_4850_pochhammer__rec,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),N)) ).

% pochhammer_rec
tff(fact_4851_pochhammer__rec,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),one_one_complex)),N)) ).

% pochhammer_rec
tff(fact_4852_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% pochhammer_Suc
tff(fact_4853_pochhammer__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(comm_s2602460028002588243omplex(A),N)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_nat_complex(semiri8010041392384452111omplex,N))) ).

% pochhammer_Suc
tff(fact_4854_pochhammer__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(A),N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% pochhammer_Suc
tff(fact_4855_pochhammer__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% pochhammer_Suc
tff(fact_4856_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% pochhammer_Suc
tff(fact_4857_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).

% pochhammer_rec'
tff(fact_4858_pochhammer__rec_H,axiom,
    ! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_nat_complex(semiri8010041392384452111omplex,N))),aa_nat_complex(comm_s2602460028002588243omplex(Z),N)) ).

% pochhammer_rec'
tff(fact_4859_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(comm_s7457072308508201937r_real(Z),N)) ).

% pochhammer_rec'
tff(fact_4860_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(comm_s4660882817536571857er_int(Z),N)) ).

% pochhammer_rec'
tff(fact_4861_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).

% pochhammer_rec'
tff(fact_4862_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,N))),M)) ).

% pochhammer_product'
tff(fact_4863_pochhammer__product_H,axiom,
    ! [Z: complex,N: nat,M: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(comm_s2602460028002588243omplex(Z),N)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_nat_complex(semiri8010041392384452111omplex,N))),M)) ).

% pochhammer_product'
tff(fact_4864_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),N)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),M)) ).

% pochhammer_product'
tff(fact_4865_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ).

% pochhammer_product'
tff(fact_4866_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ).

% pochhammer_product'
tff(fact_4867_subset__Collect__iff,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B3),A2))
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B3),collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bg(A2),P))))
      <=> ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),B3))
           => pp(aa_VEBT_VEBT_bool(P,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_4868_subset__Collect__iff,axiom,
    ! [B3: set_real,A2: set_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(B3),A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(B3),collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bh(A2),P))))
      <=> ! [X4: real] :
            ( pp(aa_set_real_bool(member_real(X4),B3))
           => pp(aa_real_bool(P,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_4869_subset__Collect__iff,axiom,
    ! [B3: set_complex,A2: set_complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B3),A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B3),collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_bi(A2),P))))
      <=> ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),B3))
           => pp(aa_complex_bool(P,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_4870_subset__Collect__iff,axiom,
    ! [B3: set_list_nat,A2: set_list_nat,P: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(B3,A2)
     => ( ord_le6045566169113846134st_nat(B3,collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bj(A2),P)))
      <=> ! [X4: list_nat] :
            ( pp(aa_set_list_nat_bool(member_list_nat(X4),B3))
           => pp(aa_list_nat_bool(P,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_4871_subset__Collect__iff,axiom,
    ! [B3: set_set_nat,A2: set_set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(B3),A2))
     => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(B3),collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bk(A2),P))))
      <=> ! [X4: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X4),B3))
           => pp(aa_set_nat_bool(P,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_4872_subset__Collect__iff,axiom,
    ! [B3: set_nat,A2: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bl(A2),P))))
      <=> ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),B3))
           => pp(aa_nat_bool(P,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_4873_subset__Collect__iff,axiom,
    ! [B3: set_int,A2: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bm(A2),P))))
      <=> ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),B3))
           => pp(aa_int_bool(P,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_4874_subset__CollectI,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,Q: fun_VEBT_VEBT_bool,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(B3),A2))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),B3))
           => ( pp(aa_VEBT_VEBT_bool(Q,X))
             => pp(aa_VEBT_VEBT_bool(P,X)) ) )
       => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bg(B3),Q))),collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bg(A2),P)))) ) ) ).

% subset_CollectI
tff(fact_4875_subset__CollectI,axiom,
    ! [B3: set_real,A2: set_real,Q: fun_real_bool,P: fun_real_bool] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(B3),A2))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),B3))
           => ( pp(aa_real_bool(Q,X))
             => pp(aa_real_bool(P,X)) ) )
       => pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bh(B3),Q))),collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bh(A2),P)))) ) ) ).

% subset_CollectI
tff(fact_4876_subset__CollectI,axiom,
    ! [B3: set_complex,A2: set_complex,Q: fun_complex_bool,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B3),A2))
     => ( ! [X: complex] :
            ( pp(aa_set_complex_bool(member_complex(X),B3))
           => ( pp(aa_complex_bool(Q,X))
             => pp(aa_complex_bool(P,X)) ) )
       => pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_bi(B3),Q))),collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_bi(A2),P)))) ) ) ).

% subset_CollectI
tff(fact_4877_subset__CollectI,axiom,
    ! [B3: set_list_nat,A2: set_list_nat,Q: fun_list_nat_bool,P: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(B3,A2)
     => ( ! [X: list_nat] :
            ( pp(aa_set_list_nat_bool(member_list_nat(X),B3))
           => ( pp(aa_list_nat_bool(Q,X))
             => pp(aa_list_nat_bool(P,X)) ) )
       => ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bj(B3),Q)),collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_bj(A2),P))) ) ) ).

% subset_CollectI
tff(fact_4878_subset__CollectI,axiom,
    ! [B3: set_set_nat,A2: set_set_nat,Q: fun_set_nat_bool,P: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(B3),A2))
     => ( ! [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),B3))
           => ( pp(aa_set_nat_bool(Q,X))
             => pp(aa_set_nat_bool(P,X)) ) )
       => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bk(B3),Q))),collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_bk(A2),P)))) ) ) ).

% subset_CollectI
tff(fact_4879_subset__CollectI,axiom,
    ! [B3: set_nat,A2: set_nat,Q: fun_nat_bool,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),B3))
           => ( pp(aa_nat_bool(Q,X))
             => pp(aa_nat_bool(P,X)) ) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bl(B3),Q))),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bl(A2),P)))) ) ) ).

% subset_CollectI
tff(fact_4880_subset__CollectI,axiom,
    ! [B3: set_int,A2: set_int,Q: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),B3))
           => ( pp(aa_int_bool(Q,X))
             => pp(aa_int_bool(P,X)) ) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bm(B3),Q))),collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bm(A2),P)))) ) ) ).

% subset_CollectI
tff(fact_4881_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4882_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z),N) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(comm_s2602460028002588243omplex(Z),M)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_nat_complex(semiri8010041392384452111omplex,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4883_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_real(comm_s7457072308508201937r_real(Z),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z),M)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4884_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4885_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% pochhammer_product
tff(fact_4886_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_4887_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),X3))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = one_one_real )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = 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)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_4888_Tbuildupi__buildupi_H,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,N)) = aa_nat_int(vEBT_V9176841429113362141ildupi,N) ).

% Tbuildupi_buildupi'
tff(fact_4889_Bolzano,axiom,
    ! [A: real,B: real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [A3: real,B2: real,C4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B2),C4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),B2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),C4))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),C4)) ) ) ) )
       => ( ! [X: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
               => ? [D3: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
                    & ! [A3: real,B2: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),X))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B2))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(B2),A3)),D3)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B2)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% Bolzano
tff(fact_4890_Set__filter__fold,axiom,
    ! [A2: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( filter_VEBT_VEBT(P,A2) = finite2755337914281901096T_VEBT(aTP_Lamm_rl(P),bot_bo8194388402131092736T_VEBT,A2) ) ) ).

% Set_filter_fold
tff(fact_4891_Set__filter__fold,axiom,
    ! [A2: set_complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( filter_complex(P,A2) = finite7687009701937157724omplex(aTP_Lamm_rm(P),bot_bot_set_complex,A2) ) ) ).

% Set_filter_fold
tff(fact_4892_Set__filter__fold,axiom,
    ! [A2: set_Code_integer,P: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( filter_Code_integer(P,A2) = finite7486036041278456378nteger(aTP_Lamm_rn(P),bot_bo3990330152332043303nteger,A2) ) ) ).

% Set_filter_fold
tff(fact_4893_Set__filter__fold,axiom,
    ! [A2: set_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( filter_real(P,A2) = finite3596656892938901080t_real(aTP_Lamm_ro(P),bot_bot_set_real,A2) ) ) ).

% Set_filter_fold
tff(fact_4894_Set__filter__fold,axiom,
    ! [A2: set_bool,P: fun_bool_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( filter_bool(P,A2) = finite3373824251209765736t_bool(aTP_Lamm_rp(P),bot_bot_set_bool,A2) ) ) ).

% Set_filter_fold
tff(fact_4895_Set__filter__fold,axiom,
    ! [A2: set_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_set_nat(filter_nat2(P),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(finite5529483035118572448et_nat(aTP_Lamm_rq(P)),bot_bot_set_nat),A2) ) ) ).

% Set_filter_fold
tff(fact_4896_Set__filter__fold,axiom,
    ! [A2: set_int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( filter_int(P,A2) = aa_set_int_set_int(aa_set4234024024286713313et_int(finite1723285575846220376et_int(aTP_Lamm_rr(P)),bot_bot_set_int),A2) ) ) ).

% Set_filter_fold
tff(fact_4897_graph__map__upd,axiom,
    ! [M: fun_int_option_int,K: int,V: int] : graph_int_int(fun_up8666045135305973159on_int(M,K,aa_int_option_int(some_int,V))) = aa_set6728540191533482748nt_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(K),V)),graph_int_int(fun_up8666045135305973159on_int(M,K,none_int))) ).

% graph_map_upd
tff(fact_4898_graph__map__upd,axiom,
    ! [M: fun_Co8029881127924365703nteger,K: code_integer,V: code_integer] : graph_5282091004195177018nteger(fun_up6566892301432185865nteger(M,K,aa_Cod5017737808611256136nteger(some_Code_integer,V))) = aa_set6745602309116264572nteger(insert4913895101485356395nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(K),V)),graph_5282091004195177018nteger(fun_up6566892301432185865nteger(M,K,none_Code_integer))) ).

% graph_map_upd
tff(fact_4899_graph__map__upd,axiom,
    ! [M: fun_nat_option_nat,K: nat,V: nat] : graph_nat_nat(fun_up1493157387958331631on_nat(M,K,aa_nat_option_nat(some_nat,V))) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),V)),graph_nat_nat(fun_up1493157387958331631on_nat(M,K,none_nat))) ).

% graph_map_upd
tff(fact_4900_graph__map__upd,axiom,
    ! [M: fun_VE8174548813682830439on_nat,K: vEBT_VEBT,V: nat] : graph_VEBT_VEBT_nat(fun_up5885881570350532375on_nat(M,K,aa_nat_option_nat(some_nat,V))) = aa_set1503868022174493200BT_nat(insert8978894354669351395BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(K),V)),graph_VEBT_VEBT_nat(fun_up5885881570350532375on_nat(M,K,none_nat))) ).

% graph_map_upd
tff(fact_4901_graph__map__upd,axiom,
    ! [M: fun_nat_option_num,K: nat,V: num] : graph_nat_num(fun_up2201401324907169337on_num(M,K,aa_num_option_num(some_num,V))) = aa_set8037964202635646076at_num(insert8920054152555992091at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(K),V)),graph_nat_num(fun_up2201401324907169337on_num(M,K,none_num))) ).

% graph_map_upd
tff(fact_4902_finite__interval__int1,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_rs(A),B)))) ).

% finite_interval_int1
tff(fact_4903_finite__interval__int4,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_rt(A),B)))) ).

% finite_interval_int4
tff(fact_4904_sup__Some,axiom,
    ! [X3: assn,Y: assn] : sup_sup_option_assn(some_assn(X3),some_assn(Y)) = some_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y)) ).

% sup_Some
tff(fact_4905_sup__Some,axiom,
    ! [X3: set_nat,Y: set_nat] : sup_su3598758113090618626et_nat(aa_set6119337032432945050et_nat(some_set_nat,X3),aa_set6119337032432945050et_nat(some_set_nat,Y)) = aa_set6119337032432945050et_nat(some_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y)) ).

% sup_Some
tff(fact_4906_sup__Some,axiom,
    ! [X3: nat,Y: nat] : sup_sup_option_nat(aa_nat_option_nat(some_nat,X3),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y)) ).

% sup_Some
tff(fact_4907_Un__empty,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),B3) = bot_bot_set_real )
    <=> ( ( A2 = bot_bot_set_real )
        & ( B3 = bot_bot_set_real ) ) ) ).

% Un_empty
tff(fact_4908_Un__empty,axiom,
    ! [A2: set_bool,B3: set_bool] :
      ( ( aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),B3) = bot_bot_set_bool )
    <=> ( ( A2 = bot_bot_set_bool )
        & ( B3 = bot_bot_set_bool ) ) ) ).

% Un_empty
tff(fact_4909_Un__empty,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = bot_bot_set_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B3 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_4910_Un__empty,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = bot_bot_set_int )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B3 = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_4911_Un__subset__iff,axiom,
    ! [A2: set_nat,B3: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3)),C2))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C2))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),C2)) ) ) ).

% Un_subset_iff
tff(fact_4912_Un__subset__iff,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3)),C2))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C2))
        & pp(aa_set_int_bool(ord_less_eq_set_int(B3),C2)) ) ) ).

% Un_subset_iff
tff(fact_4913_finite__interval__int2,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ru(A),B)))) ).

% finite_interval_int2
tff(fact_4914_finite__interval__int3,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_rv(A),B)))) ).

% finite_interval_int3
tff(fact_4915_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ).

% zle_add1_eq_le
tff(fact_4916_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% zle_diff1_eq
tff(fact_4917_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z))
    <=> ? [N6: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N6)) ) ).

% zle_iff_zadd
tff(fact_4918_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_4919_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_4920_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_4921_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_4922_pos__mult__pos__ge,axiom,
    ! [X3: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,N),X3))) ) ) ).

% pos_mult_pos_ge
tff(fact_4923_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q5)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_4924_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q4)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_4925_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q4)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_4926_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B5))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_4927_q__pos__lemma,axiom,
    ! [B5: int,Q5: int,R4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q5)) ) ) ) ).

% q_pos_lemma
tff(fact_4928_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z))) ) ).

% le_imp_0_less
tff(fact_4929_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z)) ) ).

% zless_imp_add1_zle
tff(fact_4930_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ).

% add1_zle_eq
tff(fact_4931_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X: int] :
            ( pp(aa_int_bool(P,X))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X5: int] :
              ( pp(aa_int_bool(P,X5))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_4932_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
           => ( pp(aa_int_bool(P,I2))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_4933_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_4934_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X: int] :
            ( pp(aa_int_bool(P,X))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X5: int] :
              ( pp(aa_int_bool(P,X5))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4935_less__1__helper,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int)),M)) ) ).

% less_1_helper
tff(fact_4936_minusinfinity,axiom,
    ! [D: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X: int,K2: int] :
            ( pp(aa_int_bool(P1,X))
          <=> pp(aa_int_bool(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] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z4))
             => ( pp(aa_int_bool(P,X))
              <=> pp(aa_int_bool(P1,X)) ) )
         => ( ? [X_1: int] : pp(aa_int_bool(P1,X_1))
           => ? [X_13: int] : pp(aa_int_bool(P,X_13)) ) ) ) ) ).

% minusinfinity
tff(fact_4937_plusinfinity,axiom,
    ! [D: int,P2: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X: int,K2: int] :
            ( pp(aa_int_bool(P2,X))
          <=> pp(aa_int_bool(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)))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X))
             => ( pp(aa_int_bool(P,X))
              <=> pp(aa_int_bool(P2,X)) ) )
         => ( ? [X_1: int] : pp(aa_int_bool(P2,X_1))
           => ? [X_13: int] : pp(aa_int_bool(P,X_13)) ) ) ) ) ).

% plusinfinity
tff(fact_4938_odd__nonzero,axiom,
    ! [Z: 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 ).

% odd_nonzero
tff(fact_4939_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% odd_less_0_iff
tff(fact_4940_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_4941_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_4942_less__int__code_I1_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_int_code(1)
tff(fact_4943_Collect__disj__eq,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] : collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_cv(P),Q)) = sup_sup_set_complex(collect_complex(P),collect_complex(Q)) ).

% Collect_disj_eq
tff(fact_4944_Collect__disj__eq,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] : collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_cr(P),Q)) = sup_sup_set_list_nat(collect_list_nat(P),collect_list_nat(Q)) ).

% Collect_disj_eq
tff(fact_4945_Collect__disj__eq,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] : collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_cs(P),Q)) = sup_sup_set_set_nat(collect_set_nat(P),collect_set_nat(Q)) ).

% Collect_disj_eq
tff(fact_4946_Collect__disj__eq,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] : collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_cu(P),Q)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,collect_int(P)),collect_int(Q)) ).

% Collect_disj_eq
tff(fact_4947_Collect__disj__eq,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] : collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ct(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(P)),collect_nat(Q)) ).

% Collect_disj_eq
tff(fact_4948_Un__def,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] : sup_su6272177626956685416T_VEBT(A2,B3) = collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_rw(A2),B3)) ).

% Un_def
tff(fact_4949_Un__def,axiom,
    ! [A2: set_real,B3: set_real] : aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),B3) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_rx(A2),B3)) ).

% Un_def
tff(fact_4950_Un__def,axiom,
    ! [A2: set_complex,B3: set_complex] : sup_sup_set_complex(A2,B3) = collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_ry(A2),B3)) ).

% Un_def
tff(fact_4951_Un__def,axiom,
    ! [A2: set_list_nat,B3: set_list_nat] : sup_sup_set_list_nat(A2,B3) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_rz(A2),B3)) ).

% Un_def
tff(fact_4952_Un__def,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] : sup_sup_set_set_nat(A2,B3) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_sa(A2),B3)) ).

% Un_def
tff(fact_4953_Un__def,axiom,
    ! [A2: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_sb(A2),B3)) ).

% Un_def
tff(fact_4954_Un__def,axiom,
    ! [A2: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_sc(A2),B3)) ).

% Un_def
tff(fact_4955_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_4956_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_4957_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_4958_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_4959_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_4960_less__eq__int__code_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% less_eq_int_code(1)
tff(fact_4961_imp__le__cong,axiom,
    ! [X3: int,X7: int,P: bool,P2: bool] :
      ( ( X3 = X7 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X7))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
           => pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X7))
           => pp(P2) ) ) ) ) ).

% imp_le_cong
tff(fact_4962_conj__le__cong,axiom,
    ! [X3: int,X7: int,P: bool,P2: bool] :
      ( ( X3 = X7 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X7))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
            & pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X7))
            & pp(P2) ) ) ) ) ).

% conj_le_cong
tff(fact_4963_verit__la__generic,axiom,
    ! [A: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X3))
      | ( A = X3 )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),A)) ) ).

% verit_la_generic
tff(fact_4964_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(set_or4662586982721622107an_int(zero_zero_int),U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_4965_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),bot_bot_assn) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4966_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_real] : aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,X3),bot_bot_set_real) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4967_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,X3),bot_bot_set_bool) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4968_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),bot_bot_set_nat) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4969_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),bot_bot_set_int) = X3 ).

% boolean_algebra.disj_zero_right
tff(fact_4970_foldl__un__empty__eq,axiom,
    ! [I: set_real,Ww: list_set_real] : foldl_4142838678319929319t_real(sup_sup_set_real,I,Ww) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,I),foldl_4142838678319929319t_real(sup_sup_set_real,bot_bot_set_real,Ww)) ).

% foldl_un_empty_eq
tff(fact_4971_foldl__un__empty__eq,axiom,
    ! [I: set_bool,Ww: list_set_bool] : foldl_3150261563131881825t_bool(sup_sup_set_bool,I,Ww) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,I),foldl_3150261563131881825t_bool(sup_sup_set_bool,bot_bot_set_bool,Ww)) ).

% foldl_un_empty_eq
tff(fact_4972_foldl__un__empty__eq,axiom,
    ! [I: set_nat,Ww: list_set_nat] : foldl_4988731653086973103et_nat(sup_sup_set_nat,I,Ww) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,I),foldl_4988731653086973103et_nat(sup_sup_set_nat,bot_bot_set_nat,Ww)) ).

% foldl_un_empty_eq
tff(fact_4973_foldl__un__empty__eq,axiom,
    ! [I: set_int,Ww: list_set_int] : foldl_6819690284573351271et_int(sup_sup_set_int,I,Ww) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,I),foldl_6819690284573351271et_int(sup_sup_set_int,bot_bot_set_int,Ww)) ).

% foldl_un_empty_eq
tff(fact_4974_Un__empty__left,axiom,
    ! [B3: set_real] : aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,bot_bot_set_real),B3) = B3 ).

% Un_empty_left
tff(fact_4975_Un__empty__left,axiom,
    ! [B3: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,bot_bot_set_bool),B3) = B3 ).

% Un_empty_left
tff(fact_4976_Un__empty__left,axiom,
    ! [B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B3) = B3 ).

% Un_empty_left
tff(fact_4977_Un__empty__left,axiom,
    ! [B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),B3) = B3 ).

% Un_empty_left
tff(fact_4978_Un__empty__right,axiom,
    ! [A2: set_real] : aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),bot_bot_set_real) = A2 ).

% Un_empty_right
tff(fact_4979_Un__empty__right,axiom,
    ! [A2: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),bot_bot_set_bool) = A2 ).

% Un_empty_right
tff(fact_4980_Un__empty__right,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),bot_bot_set_nat) = A2 ).

% Un_empty_right
tff(fact_4981_Un__empty__right,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),bot_bot_set_int) = A2 ).

% Un_empty_right
tff(fact_4982_Un__mono,axiom,
    ! [A2: set_nat,C2: set_nat,B3: set_nat,D4: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),D4))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D4))) ) ) ).

% Un_mono
tff(fact_4983_Un__mono,axiom,
    ! [A2: set_int,C2: set_int,B3: set_int,D4: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),D4))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C2),D4))) ) ) ).

% Un_mono
tff(fact_4984_Un__least,axiom,
    ! [A2: set_nat,C2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),C2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),C2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3)),C2)) ) ) ).

% Un_least
tff(fact_4985_Un__least,axiom,
    ! [A2: set_int,C2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),C2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3)),C2)) ) ) ).

% Un_least
tff(fact_4986_Un__upper1,axiom,
    ! [A2: set_nat,B3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ).

% Un_upper1
tff(fact_4987_Un__upper1,axiom,
    ! [A2: set_int,B3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3))) ).

% Un_upper1
tff(fact_4988_Un__upper2,axiom,
    ! [B3: set_nat,A2: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) ).

% Un_upper2
tff(fact_4989_Un__upper2,axiom,
    ! [B3: set_int,A2: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(B3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3))) ).

% Un_upper2
tff(fact_4990_Un__absorb1,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_4991_Un__absorb1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_4992_Un__absorb2,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = A2 ) ) ).

% Un_absorb2
tff(fact_4993_Un__absorb2,axiom,
    ! [B3: set_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = A2 ) ) ).

% Un_absorb2
tff(fact_4994_subset__UnE,axiom,
    ! [C2: set_nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3)))
     => ~ ! [A9: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A9),A2))
           => ! [B6: set_nat] :
                ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B6),B3))
               => ( C2 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A9),B6) ) ) ) ) ).

% subset_UnE
tff(fact_4995_subset__UnE,axiom,
    ! [C2: set_int,A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3)))
     => ~ ! [A9: set_int] :
            ( pp(aa_set_int_bool(ord_less_eq_set_int(A9),A2))
           => ! [B6: set_int] :
                ( pp(aa_set_int_bool(ord_less_eq_set_int(B6),B3))
               => ( C2 != aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A9),B6) ) ) ) ) ).

% subset_UnE
tff(fact_4996_subset__Un__eq,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_4997_subset__Un__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_4998_Set_Ofilter__def,axiom,
    ! [P: fun_VEBT_VEBT_bool,A2: set_VEBT_VEBT] : filter_VEBT_VEBT(P,A2) = collect_VEBT_VEBT(aa_set1728193350978918526T_bool(aTP_Lamm_sd(P),A2)) ).

% Set.filter_def
tff(fact_4999_Set_Ofilter__def,axiom,
    ! [P: fun_real_bool,A2: set_real] : filter_real(P,A2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_se(P),A2)) ).

% Set.filter_def
tff(fact_5000_Set_Ofilter__def,axiom,
    ! [P: fun_complex_bool,A2: set_complex] : filter_complex(P,A2) = collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_sf(P),A2)) ).

% Set.filter_def
tff(fact_5001_Set_Ofilter__def,axiom,
    ! [P: fun_list_nat_bool,A2: set_list_nat] : filter_list_nat(P,A2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_sg(P),A2)) ).

% Set.filter_def
tff(fact_5002_Set_Ofilter__def,axiom,
    ! [P: fun_set_nat_bool,A2: set_set_nat] : filter_set_nat(P,A2) = collect_set_nat(aa_set9121396911425987952t_bool(aTP_Lamm_sh(P),A2)) ).

% Set.filter_def
tff(fact_5003_Set_Ofilter__def,axiom,
    ! [P: fun_nat_bool,A2: set_nat] : aa_set_nat_set_nat(filter_nat2(P),A2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_si(P),A2)) ).

% Set.filter_def
tff(fact_5004_Set_Ofilter__def,axiom,
    ! [P: fun_int_bool,A2: set_int] : filter_int(P,A2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_sj(P),A2)) ).

% Set.filter_def
tff(fact_5005_insert__def,axiom,
    ! [A: vEBT_VEBT,B3: set_VEBT_VEBT] : aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B3) = sup_su6272177626956685416T_VEBT(collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aTP_Lamm_je,A)),B3) ).

% insert_def
tff(fact_5006_insert__def,axiom,
    ! [A: bool,B3: set_bool] : aa_set_bool_set_bool(insert_bool(A),B3) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,collect_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_jj,A))),B3) ).

% insert_def
tff(fact_5007_insert__def,axiom,
    ! [A: real,B3: set_real] : aa_set_real_set_real(insert_real(A),B3) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ji,A))),B3) ).

% insert_def
tff(fact_5008_insert__def,axiom,
    ! [A: complex,B3: set_complex] : insert_complex(A,B3) = sup_sup_set_complex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_jf,A)),B3) ).

% insert_def
tff(fact_5009_insert__def,axiom,
    ! [A: list_nat,B3: set_list_nat] : insert_list_nat(A,B3) = sup_sup_set_list_nat(collect_list_nat(aa_lis4768068104879709938t_bool(aTP_Lamm_jg,A)),B3) ).

% insert_def
tff(fact_5010_insert__def,axiom,
    ! [A: set_nat,B3: set_set_nat] : aa_set1013257548857649334et_nat(insert_set_nat(A),B3) = sup_sup_set_set_nat(collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_jh,A)),B3) ).

% insert_def
tff(fact_5011_insert__def,axiom,
    ! [A: int,B3: set_int] : aa_set_int_set_int(insert_int(A),B3) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_jl,A))),B3) ).

% insert_def
tff(fact_5012_insert__def,axiom,
    ! [A: nat,B3: set_nat] : aa_set_nat_set_nat(insert_nat(A),B3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_jk,A))),B3) ).

% insert_def
tff(fact_5013_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_5014_insert__is__Un,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] : aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A2) = sup_su6272177626956685416T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT),A2) ).

% insert_is_Un
tff(fact_5015_insert__is__Un,axiom,
    ! [A: real,A2: set_real] : aa_set_real_set_real(insert_real(A),A2) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,aa_set_real_set_real(insert_real(A),bot_bot_set_real)),A2) ).

% insert_is_Un
tff(fact_5016_insert__is__Un,axiom,
    ! [A: bool,A2: set_bool] : aa_set_bool_set_bool(insert_bool(A),A2) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool)),A2) ).

% insert_is_Un
tff(fact_5017_insert__is__Un,axiom,
    ! [A: nat,A2: set_nat] : aa_set_nat_set_nat(insert_nat(A),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)),A2) ).

% insert_is_Un
tff(fact_5018_insert__is__Un,axiom,
    ! [A: int,A2: set_int] : aa_set_int_set_int(insert_int(A),A2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(insert_int(A),bot_bot_set_int)),A2) ).

% insert_is_Un
tff(fact_5019_Un__singleton__iff,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( sup_su6272177626956685416T_VEBT(A2,B3) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) )
    <=> ( ( ( A2 = bot_bo8194388402131092736T_VEBT )
          & ( B3 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) ) )
        | ( ( A2 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) )
          & ( B3 = bot_bo8194388402131092736T_VEBT ) )
        | ( ( A2 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) )
          & ( B3 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) ) ) ) ) ).

% Un_singleton_iff
tff(fact_5020_Un__singleton__iff,axiom,
    ! [A2: set_real,B3: set_real,X3: real] :
      ( ( aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),B3) = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) )
    <=> ( ( ( A2 = bot_bot_set_real )
          & ( B3 = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) ) )
        | ( ( A2 = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) )
          & ( B3 = bot_bot_set_real ) )
        | ( ( A2 = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) )
          & ( B3 = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) ) ) ) ) ).

% Un_singleton_iff
tff(fact_5021_Un__singleton__iff,axiom,
    ! [A2: set_bool,B3: set_bool,X3: bool] :
      ( ( aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),B3) = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) )
    <=> ( ( ( A2 = bot_bot_set_bool )
          & ( B3 = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) ) )
        | ( ( A2 = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) )
          & ( B3 = bot_bot_set_bool ) )
        | ( ( A2 = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) )
          & ( B3 = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) ) ) ) ) ).

% Un_singleton_iff
tff(fact_5022_Un__singleton__iff,axiom,
    ! [A2: set_nat,B3: set_nat,X3: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B3 = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ) )
        | ( ( A2 = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) )
          & ( B3 = bot_bot_set_nat ) )
        | ( ( A2 = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) )
          & ( B3 = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_5023_Un__singleton__iff,axiom,
    ! [A2: set_int,B3: set_int,X3: int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B3 = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ) )
        | ( ( A2 = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) )
          & ( B3 = bot_bot_set_int ) )
        | ( ( A2 = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) )
          & ( B3 = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_5024_singleton__Un__iff,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) = sup_su6272177626956685416T_VEBT(A2,B3) )
    <=> ( ( ( A2 = bot_bo8194388402131092736T_VEBT )
          & ( B3 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) ) )
        | ( ( A2 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) )
          & ( B3 = bot_bo8194388402131092736T_VEBT ) )
        | ( ( A2 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) )
          & ( B3 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) ) ) ) ) ).

% singleton_Un_iff
tff(fact_5025_singleton__Un__iff,axiom,
    ! [X3: real,A2: set_real,B3: set_real] :
      ( ( aa_set_real_set_real(insert_real(X3),bot_bot_set_real) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),B3) )
    <=> ( ( ( A2 = bot_bot_set_real )
          & ( B3 = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) ) )
        | ( ( A2 = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) )
          & ( B3 = bot_bot_set_real ) )
        | ( ( A2 = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) )
          & ( B3 = aa_set_real_set_real(insert_real(X3),bot_bot_set_real) ) ) ) ) ).

% singleton_Un_iff
tff(fact_5026_singleton__Un__iff,axiom,
    ! [X3: bool,A2: set_bool,B3: set_bool] :
      ( ( aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),B3) )
    <=> ( ( ( A2 = bot_bot_set_bool )
          & ( B3 = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) ) )
        | ( ( A2 = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) )
          & ( B3 = bot_bot_set_bool ) )
        | ( ( A2 = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) )
          & ( B3 = aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool) ) ) ) ) ).

% singleton_Un_iff
tff(fact_5027_singleton__Un__iff,axiom,
    ! [X3: nat,A2: set_nat,B3: set_nat] :
      ( ( aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B3 = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ) )
        | ( ( A2 = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) )
          & ( B3 = bot_bot_set_nat ) )
        | ( ( A2 = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) )
          & ( B3 = aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_5028_singleton__Un__iff,axiom,
    ! [X3: int,A2: set_int,B3: set_int] :
      ( ( aa_set_int_set_int(insert_int(X3),bot_bot_set_int) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B3 = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ) )
        | ( ( A2 = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) )
          & ( B3 = bot_bot_set_int ) )
        | ( ( A2 = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) )
          & ( B3 = aa_set_int_set_int(insert_int(X3),bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_5029_ivl__disj__un__two_I3_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),M))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,M),U))
       => ( sup_sup_set_rat(set_or4029947393144176647an_rat(L,M),set_or4029947393144176647an_rat(M,U)) = set_or4029947393144176647an_rat(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_5030_ivl__disj__un__two_I3_J,axiom,
    ! [L: num,M: num,U: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),M))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),U))
       => ( sup_sup_set_num(set_or1222409239386451017an_num(L,M),set_or1222409239386451017an_num(M,U)) = set_or1222409239386451017an_num(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_5031_ivl__disj__un__two_I3_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),U))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),U)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_5032_ivl__disj__un__two_I3_J,axiom,
    ! [L: int,M: int,U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),U))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(set_or4662586982721622107an_int(L),M)),aa_int_set_int(set_or4662586982721622107an_int(M),U)) = aa_int_set_int(set_or4662586982721622107an_int(L),U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_5033_ivl__disj__un__two_I3_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,L),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,M),U))
       => ( sup_su848401254843788991nteger(set_or8404916559141939852nteger(L,M),set_or8404916559141939852nteger(M,U)) = set_or8404916559141939852nteger(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_5034_Diff__subset__conv,axiom,
    ! [A2: set_nat,B3: set_nat,C2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3)),C2))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B3),C2))) ) ).

% Diff_subset_conv
tff(fact_5035_Diff__subset__conv,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B3)),C2))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B3),C2))) ) ).

% Diff_subset_conv
tff(fact_5036_Diff__partition,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B3),A2)) = B3 ) ) ).

% Diff_partition
tff(fact_5037_Diff__partition,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B3),A2)) = B3 ) ) ).

% Diff_partition
tff(fact_5038_in__graphD,axiom,
    ! [K: int,V: int,M: fun_int_option_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(K),V),graph_int_int(M))
     => ( aa_int_option_int(M,K) = aa_int_option_int(some_int,V) ) ) ).

% in_graphD
tff(fact_5039_in__graphD,axiom,
    ! [K: code_integer,V: code_integer,M: fun_Co8029881127924365703nteger] :
      ( member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(K),V),graph_5282091004195177018nteger(M))
     => ( aa_Cod5017737808611256136nteger(M,K) = aa_Cod5017737808611256136nteger(some_Code_integer,V) ) ) ).

% in_graphD
tff(fact_5040_in__graphD,axiom,
    ! [K: nat,V: nat,M: fun_nat_option_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),V),graph_nat_nat(M))
     => ( aa_nat_option_nat(M,K) = aa_nat_option_nat(some_nat,V) ) ) ).

% in_graphD
tff(fact_5041_in__graphD,axiom,
    ! [K: vEBT_VEBT,V: nat,M: fun_VE8174548813682830439on_nat] :
      ( member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(K),V),graph_VEBT_VEBT_nat(M))
     => ( aa_VEB7679222860772546456on_nat(M,K) = aa_nat_option_nat(some_nat,V) ) ) ).

% in_graphD
tff(fact_5042_in__graphD,axiom,
    ! [K: nat,V: num,M: fun_nat_option_num] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(K),V),graph_nat_num(M))
     => ( aa_nat_option_num(M,K) = aa_num_option_num(some_num,V) ) ) ).

% in_graphD
tff(fact_5043_in__graphI,axiom,
    ! [M: fun_int_option_int,K: int,V: int] :
      ( ( aa_int_option_int(M,K) = aa_int_option_int(some_int,V) )
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(K),V),graph_int_int(M)) ) ).

% in_graphI
tff(fact_5044_in__graphI,axiom,
    ! [M: fun_Co8029881127924365703nteger,K: code_integer,V: code_integer] :
      ( ( aa_Cod5017737808611256136nteger(M,K) = aa_Cod5017737808611256136nteger(some_Code_integer,V) )
     => member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(K),V),graph_5282091004195177018nteger(M)) ) ).

% in_graphI
tff(fact_5045_in__graphI,axiom,
    ! [M: fun_nat_option_nat,K: nat,V: nat] :
      ( ( aa_nat_option_nat(M,K) = aa_nat_option_nat(some_nat,V) )
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),V),graph_nat_nat(M)) ) ).

% in_graphI
tff(fact_5046_in__graphI,axiom,
    ! [M: fun_VE8174548813682830439on_nat,K: vEBT_VEBT,V: nat] :
      ( ( aa_VEB7679222860772546456on_nat(M,K) = aa_nat_option_nat(some_nat,V) )
     => member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(K),V),graph_VEBT_VEBT_nat(M)) ) ).

% in_graphI
tff(fact_5047_in__graphI,axiom,
    ! [M: fun_nat_option_num,K: nat,V: num] :
      ( ( aa_nat_option_num(M,K) = aa_num_option_num(some_num,V) )
     => member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(K),V),graph_nat_num(M)) ) ).

% in_graphI
tff(fact_5048_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_5049_sup__bot__left,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),X3) = X3 ).

% sup_bot_left
tff(fact_5050_sup__bot__left,axiom,
    ! [X3: set_real] : aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,bot_bot_set_real),X3) = X3 ).

% sup_bot_left
tff(fact_5051_sup__bot__left,axiom,
    ! [X3: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,bot_bot_set_bool),X3) = X3 ).

% sup_bot_left
tff(fact_5052_sup__bot__left,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),X3) = X3 ).

% sup_bot_left
tff(fact_5053_sup__bot__left,axiom,
    ! [X3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),X3) = X3 ).

% sup_bot_left
tff(fact_5054_sup__bot__right,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),bot_bot_assn) = X3 ).

% sup_bot_right
tff(fact_5055_sup__bot__right,axiom,
    ! [X3: set_real] : aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,X3),bot_bot_set_real) = X3 ).

% sup_bot_right
tff(fact_5056_sup__bot__right,axiom,
    ! [X3: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,X3),bot_bot_set_bool) = X3 ).

% sup_bot_right
tff(fact_5057_sup__bot__right,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),bot_bot_set_nat) = X3 ).

% sup_bot_right
tff(fact_5058_sup__bot__right,axiom,
    ! [X3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),bot_bot_set_int) = X3 ).

% sup_bot_right
tff(fact_5059_bot__eq__sup__iff,axiom,
    ! [X3: assn,Y: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y) )
    <=> ( ( X3 = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
tff(fact_5060_bot__eq__sup__iff,axiom,
    ! [X3: set_real,Y: set_real] :
      ( ( bot_bot_set_real = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,X3),Y) )
    <=> ( ( X3 = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% bot_eq_sup_iff
tff(fact_5061_bot__eq__sup__iff,axiom,
    ! [X3: set_bool,Y: set_bool] :
      ( ( bot_bot_set_bool = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,X3),Y) )
    <=> ( ( X3 = bot_bot_set_bool )
        & ( Y = bot_bot_set_bool ) ) ) ).

% bot_eq_sup_iff
tff(fact_5062_bot__eq__sup__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) )
    <=> ( ( X3 = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_5063_bot__eq__sup__iff,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y) )
    <=> ( ( X3 = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
tff(fact_5064_sup__eq__bot__iff,axiom,
    ! [X3: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y) = bot_bot_assn )
    <=> ( ( X3 = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
tff(fact_5065_sup__eq__bot__iff,axiom,
    ! [X3: set_real,Y: set_real] :
      ( ( aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,X3),Y) = bot_bot_set_real )
    <=> ( ( X3 = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% sup_eq_bot_iff
tff(fact_5066_sup__eq__bot__iff,axiom,
    ! [X3: set_bool,Y: set_bool] :
      ( ( aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,X3),Y) = bot_bot_set_bool )
    <=> ( ( X3 = bot_bot_set_bool )
        & ( Y = bot_bot_set_bool ) ) ) ).

% sup_eq_bot_iff
tff(fact_5067_sup__eq__bot__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = bot_bot_set_nat )
    <=> ( ( X3 = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_5068_sup__eq__bot__iff,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y) = bot_bot_set_int )
    <=> ( ( X3 = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
tff(fact_5069_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = bot_bot_assn )
    <=> ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_5070_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A),B) = bot_bot_set_real )
    <=> ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_5071_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ( aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A),B) = bot_bot_set_bool )
    <=> ( ( A = bot_bot_set_bool )
        & ( B = bot_bot_set_bool ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_5072_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_5073_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_5074_sup__bot_Oleft__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),A) = A ).

% sup_bot.left_neutral
tff(fact_5075_sup__bot_Oleft__neutral,axiom,
    ! [A: set_real] : aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,bot_bot_set_real),A) = A ).

% sup_bot.left_neutral
tff(fact_5076_sup__bot_Oleft__neutral,axiom,
    ! [A: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,bot_bot_set_bool),A) = A ).

% sup_bot.left_neutral
tff(fact_5077_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_5078_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),A) = A ).

% sup_bot.left_neutral
tff(fact_5079_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
    <=> ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_5080_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( bot_bot_set_real = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A),B) )
    <=> ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_5081_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ( bot_bot_set_bool = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A),B) )
    <=> ( ( A = bot_bot_set_bool )
        & ( B = bot_bot_set_bool ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_5082_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_5083_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_5084_merge__pure__or,axiom,
    ! [A: bool,B: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_bool_assn(pure_assn,A)),aa_bool_assn(pure_assn,B)) = aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,A),B)) ).

% merge_pure_or
tff(fact_5085_sup_Obounded__iff,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C),A)
    <=> ( ord_less_eq_assn(B,A)
        & ord_less_eq_assn(C,A) ) ) ).

% sup.bounded_iff
tff(fact_5086_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A)) ) ) ).

% sup.bounded_iff
tff(fact_5087_sup_Obounded__iff,axiom,
    ! [B: set_int,C: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)),A))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
        & pp(aa_set_int_bool(ord_less_eq_set_int(C),A)) ) ) ).

% sup.bounded_iff
tff(fact_5088_sup_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% sup.bounded_iff
tff(fact_5089_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% sup.bounded_iff
tff(fact_5090_sup_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% sup.bounded_iff
tff(fact_5091_le__sup__iff,axiom,
    ! [X3: assn,Y: assn,Z: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y),Z)
    <=> ( ord_less_eq_assn(X3,Z)
        & ord_less_eq_assn(Y,Z) ) ) ).

% le_sup_iff
tff(fact_5092_le__sup__iff,axiom,
    ! [X3: set_nat,Y: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y)),Z))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Z))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_5093_le__sup__iff,axiom,
    ! [X3: set_int,Y: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y)),Z))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Z))
        & pp(aa_set_int_bool(ord_less_eq_set_int(Y),Z)) ) ) ).

% le_sup_iff
tff(fact_5094_le__sup__iff,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y)),Z))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Z))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_5095_le__sup__iff,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_5096_le__sup__iff,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_5097_sup__bot_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),bot_bot_assn) = A ).

% sup_bot.right_neutral
tff(fact_5098_sup__bot_Oright__neutral,axiom,
    ! [A: set_real] : aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A),bot_bot_set_real) = A ).

% sup_bot.right_neutral
tff(fact_5099_sup__bot_Oright__neutral,axiom,
    ! [A: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A),bot_bot_set_bool) = A ).

% sup_bot.right_neutral
tff(fact_5100_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),bot_bot_set_nat) = A ).

% sup_bot.right_neutral
tff(fact_5101_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),bot_bot_set_int) = A ).

% sup_bot.right_neutral
tff(fact_5102_sup__None__1,axiom,
    ! [Y: option_nat] : sup_sup_option_nat(none_nat,Y) = Y ).

% sup_None_1
tff(fact_5103_sup__None__2,axiom,
    ! [X3: option_nat] : sup_sup_option_nat(X3,none_nat) = X3 ).

% sup_None_2
tff(fact_5104_sup__Un__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,X5: nat,Xa2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(sup_su8404041055355318397t_bool(aa_set1256819327649670340t_bool(aTP_Lamm_dp,R),aa_set1256819327649670340t_bool(aTP_Lamm_dp,S3)),X5),Xa2))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa2),sup_su6327502436637775413at_nat(R,S3)) ) ).

% sup_Un_eq2
tff(fact_5105_sup__Un__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat,X5: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(sup_su8539324196979037659t_bool(aa_set2648297115105225046t_bool(aTP_Lamm_dq,R),aa_set2648297115105225046t_bool(aTP_Lamm_dq,S3)),X5),Xa2))
    <=> member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X5),Xa2),sup_su6061789376821058069BT_nat(R,S3)) ) ).

% sup_Un_eq2
tff(fact_5106_sup__Un__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int,X5: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(sup_su8050957313097713605t_bool(aa_set8965632145344517972t_bool(aTP_Lamm_dr,R),aa_set8965632145344517972t_bool(aTP_Lamm_dr,S3)),X5),Xa2))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X5),Xa2),sup_su6024340866399070445nt_int(R,S3)) ) ).

% sup_Un_eq2
tff(fact_5107_sup__Un__eq2,axiom,
    ! [R: set_Pr4811707699266497531nteger,S3: set_Pr4811707699266497531nteger,X5: code_integer,Xa2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(sup_su4479248222115090019r_bool(aa_set8960070006189174672r_bool(aTP_Lamm_ds,R),aa_set8960070006189174672r_bool(aTP_Lamm_ds,S3)),X5),Xa2))
    <=> member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X5),Xa2),sup_su3575575879904067535nteger(R,S3)) ) ).

% sup_Un_eq2
tff(fact_5108_sup__Un__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num,X5: nat,Xa2: num] :
      ( pp(aa_num_bool(aa_nat_fun_num_bool(sup_su2374849055210163635m_bool(aa_set4109929515018738736m_bool(aTP_Lamm_dt,R),aa_set4109929515018738736m_bool(aTP_Lamm_dt,S3)),X5),Xa2))
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X5),Xa2),sup_su2042722026077122175at_num(R,S3)) ) ).

% sup_Un_eq2
tff(fact_5109_star__or__dist1,axiom,
    ! [A2: assn,B3: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B3)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B3),C2)) ).

% star_or_dist1
tff(fact_5110_star__or__dist2,axiom,
    ! [C2: assn,A2: assn,B3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B3)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),A2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),B3)) ).

% star_or_dist2
tff(fact_5111_ent__disjE,axiom,
    ! [A2: assn,C2: assn,B3: assn] :
      ( entails(A2,C2)
     => ( entails(B3,C2)
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B3),C2) ) ) ).

% ent_disjE
tff(fact_5112_ent__disjI1,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q),R)
     => entails(P,R) ) ).

% ent_disjI1
tff(fact_5113_ent__disjI2,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q),R)
     => entails(Q,R) ) ).

% ent_disjI2
tff(fact_5114_ent__disjI1_H,axiom,
    ! [A2: assn,B3: assn,C2: assn] :
      ( entails(A2,B3)
     => entails(A2,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B3),C2)) ) ).

% ent_disjI1'
tff(fact_5115_ent__disjI2_H,axiom,
    ! [A2: assn,C2: assn,B3: assn] :
      ( entails(A2,C2)
     => entails(A2,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B3),C2)) ) ).

% ent_disjI2'
tff(fact_5116_ent__disjI1__direct,axiom,
    ! [A2: assn,B3: assn] : entails(A2,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B3)) ).

% ent_disjI1_direct
tff(fact_5117_ent__disjI2__direct,axiom,
    ! [B3: assn,A2: assn] : entails(B3,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B3)) ).

% ent_disjI2_direct
tff(fact_5118_split__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,R: fun_option_nat_assn,Q: assn] :
      ( hoare_7629718768684598413on_nat(P,C,R)
     => ( hoare_7629718768684598413on_nat(Q,C,R)
       => hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q),C,R) ) ) ).

% split_rule
tff(fact_5119_split__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,R: fun_bool_assn,Q: assn] :
      ( hoare_2774773592644289642e_bool(P,C,R)
     => ( hoare_2774773592644289642e_bool(Q,C,R)
       => hoare_2774773592644289642e_bool(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q),C,R) ) ) ).

% split_rule
tff(fact_5120_split__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,R: fun_VEBT_VEBTi_assn,Q: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,R)
     => ( hoare_1429296392585015714_VEBTi(Q,C,R)
       => hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q),C,R) ) ) ).

% split_rule
tff(fact_5121_split__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,R: fun_nat_assn,Q: assn] :
      ( hoare_3067605981109127869le_nat(P,C,R)
     => ( hoare_3067605981109127869le_nat(Q,C,R)
       => hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q),C,R) ) ) ).

% split_rule
tff(fact_5122_norm__assertion__simps_I6_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),bot_bot_assn) = X3 ).

% norm_assertion_simps(6)
tff(fact_5123_norm__assertion__simps_I5_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),X3) = X3 ).

% norm_assertion_simps(5)
tff(fact_5124_sup__Un__eq,axiom,
    ! [R: set_VEBT_VEBT,S3: set_VEBT_VEBT,X5: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(sup_su7942761334433784018T_bool(aa_set1728193350978918526T_bool(aTP_Lamm_an,R),aa_set1728193350978918526T_bool(aTP_Lamm_an,S3)),X5))
    <=> pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),sup_su6272177626956685416T_VEBT(R,S3))) ) ).

% sup_Un_eq
tff(fact_5125_sup__Un__eq,axiom,
    ! [R: set_real,S3: set_real,X5: real] :
      ( pp(aa_real_bool(sup_su805534559149407256l_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ao,R),aa_set8565328311858951076l_bool(aTP_Lamm_ao,S3)),X5))
    <=> pp(aa_set_real_bool(member_real(X5),aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,R),S3))) ) ).

% sup_Un_eq
tff(fact_5126_sup__Un__eq,axiom,
    ! [R: set_int,S3: set_int,X5: int] :
      ( pp(aa_int_bool(sup_sup_fun_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_at,R),aa_set7012339369581096484t_bool(aTP_Lamm_at,S3)),X5))
    <=> pp(aa_set_int_bool(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,R),S3))) ) ).

% sup_Un_eq
tff(fact_5127_sup__Un__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat,X5: set_nat] :
      ( pp(aa_set_nat_bool(sup_su7038862894554899774t_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ar,R),aa_set9121396911425987952t_bool(aTP_Lamm_ar,S3)),X5))
    <=> pp(aa_set_set_nat_bool(member_set_nat(X5),sup_sup_set_set_nat(R,S3))) ) ).

% sup_Un_eq
tff(fact_5128_sup__Un__eq,axiom,
    ! [R: set_nat,S3: set_nat,X5: nat] :
      ( pp(aa_nat_bool(sup_sup_fun_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_as,R),aa_set3744802623701456604t_bool(aTP_Lamm_as,S3)),X5))
    <=> pp(aa_set_nat_bool(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,R),S3))) ) ).

% sup_Un_eq
tff(fact_5129_sup__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] : sup_su6272177626956685416T_VEBT(A2,B3) = collect_VEBT_VEBT(sup_su7942761334433784018T_bool(aa_set1728193350978918526T_bool(aTP_Lamm_an,A2),aa_set1728193350978918526T_bool(aTP_Lamm_an,B3))) ).

% sup_set_def
tff(fact_5130_sup__set__def,axiom,
    ! [A2: set_real,B3: set_real] : aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),B3) = collect_real(sup_su805534559149407256l_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ao,A2),aa_set8565328311858951076l_bool(aTP_Lamm_ao,B3))) ).

% sup_set_def
tff(fact_5131_sup__set__def,axiom,
    ! [A2: set_complex,B3: set_complex] : sup_sup_set_complex(A2,B3) = collect_complex(sup_su6568012871953091606x_bool(aTP_Lamm_ap(A2),aTP_Lamm_ap(B3))) ).

% sup_set_def
tff(fact_5132_sup__set__def,axiom,
    ! [A2: set_list_nat,B3: set_list_nat] : sup_sup_set_list_nat(A2,B3) = collect_list_nat(sup_su7820324534880642276t_bool(aTP_Lamm_aq(A2),aTP_Lamm_aq(B3))) ).

% sup_set_def
tff(fact_5133_sup__set__def,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] : sup_sup_set_set_nat(A2,B3) = collect_set_nat(sup_su7038862894554899774t_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ar,A2),aa_set9121396911425987952t_bool(aTP_Lamm_ar,B3))) ).

% sup_set_def
tff(fact_5134_sup__set__def,axiom,
    ! [A2: set_int,B3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3) = collect_int(sup_sup_fun_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_at,A2),aa_set7012339369581096484t_bool(aTP_Lamm_at,B3))) ).

% sup_set_def
tff(fact_5135_sup__set__def,axiom,
    ! [A2: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) = collect_nat(sup_sup_fun_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_as,A2),aa_set3744802623701456604t_bool(aTP_Lamm_as,B3))) ).

% sup_set_def
tff(fact_5136_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I),J)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_5137_if__rule__split,axiom,
    ! [B: bool,P: assn,F: heap_T2636463487746394924on_nat,Q1: fun_option_nat_assn,G: heap_T2636463487746394924on_nat,Q22: fun_option_nat_assn,Q: fun_option_nat_assn] :
      ( ( pp(B)
       => hoare_7629718768684598413on_nat(P,F,Q1) )
     => ( ( ~ pp(B)
         => hoare_7629718768684598413on_nat(P,G,Q22) )
       => ( ! [X: option_nat] : entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_option_nat_assn(Q1,X)),aa_bool_assn(pure_assn,B))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_option_nat_assn(Q22,X)),aa_bool_assn(pure_assn,aa_bool_bool(fNot,B)))),aa_option_nat_assn(Q,X))
         => hoare_7629718768684598413on_nat(P,if_Hea5867803462524415986on_nat(B,F,G),Q) ) ) ) ).

% if_rule_split
tff(fact_5138_if__rule__split,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_bool,Q1: fun_bool_assn,G: heap_Time_Heap_bool,Q22: fun_bool_assn,Q: fun_bool_assn] :
      ( ( pp(B)
       => hoare_2774773592644289642e_bool(P,F,Q1) )
     => ( ( ~ pp(B)
         => hoare_2774773592644289642e_bool(P,G,Q22) )
       => ( ! [X: bool] : entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(Q1,X)),aa_bool_assn(pure_assn,B))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(Q22,X)),aa_bool_assn(pure_assn,aa_bool_bool(fNot,B)))),aa_bool_assn(Q,X))
         => hoare_2774773592644289642e_bool(P,if_Hea7696252282596813915p_bool(B,F,G),Q) ) ) ) ).

% if_rule_split
tff(fact_5139_if__rule__split,axiom,
    ! [B: bool,P: assn,F: heap_T8145700208782473153_VEBTi,Q1: fun_VEBT_VEBTi_assn,G: heap_T8145700208782473153_VEBTi,Q22: fun_VEBT_VEBTi_assn,Q: fun_VEBT_VEBTi_assn] :
      ( ( pp(B)
       => hoare_1429296392585015714_VEBTi(P,F,Q1) )
     => ( ( ~ pp(B)
         => hoare_1429296392585015714_VEBTi(P,G,Q22) )
       => ( ! [X: vEBT_VEBTi] : entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(Q1,X)),aa_bool_assn(pure_assn,B))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(Q22,X)),aa_bool_assn(pure_assn,aa_bool_bool(fNot,B)))),aa_VEBT_VEBTi_assn(Q,X))
         => hoare_1429296392585015714_VEBTi(P,if_Hea8453224502484754311_VEBTi(B,F,G),Q) ) ) ) ).

% if_rule_split
tff(fact_5140_if__rule__split,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_nat,Q1: fun_nat_assn,G: heap_Time_Heap_nat,Q22: fun_nat_assn,Q: fun_nat_assn] :
      ( ( pp(B)
       => hoare_3067605981109127869le_nat(P,F,Q1) )
     => ( ( ~ pp(B)
         => hoare_3067605981109127869le_nat(P,G,Q22) )
       => ( ! [X: nat] : entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Q1,X)),aa_bool_assn(pure_assn,B))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Q22,X)),aa_bool_assn(pure_assn,aa_bool_bool(fNot,B)))),aa_nat_assn(Q,X))
         => hoare_3067605981109127869le_nat(P,if_Hea2662716070787841314ap_nat(B,F,G),Q) ) ) ) ).

% if_rule_split
tff(fact_5141_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X3: assn] : ord_less_eq_assn(Y,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y)) ).

% inf_sup_ord(4)
tff(fact_5142_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y))) ).

% inf_sup_ord(4)
tff(fact_5143_inf__sup__ord_I4_J,axiom,
    ! [Y: set_int,X3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(Y),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y))) ).

% inf_sup_ord(4)
tff(fact_5144_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y))) ).

% inf_sup_ord(4)
tff(fact_5145_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y))) ).

% inf_sup_ord(4)
tff(fact_5146_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y))) ).

% inf_sup_ord(4)
tff(fact_5147_inf__sup__ord_I3_J,axiom,
    ! [X3: assn,Y: assn] : ord_less_eq_assn(X3,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y)) ).

% inf_sup_ord(3)
tff(fact_5148_inf__sup__ord_I3_J,axiom,
    ! [X3: set_nat,Y: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y))) ).

% inf_sup_ord(3)
tff(fact_5149_inf__sup__ord_I3_J,axiom,
    ! [X3: set_int,Y: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y))) ).

% inf_sup_ord(3)
tff(fact_5150_inf__sup__ord_I3_J,axiom,
    ! [X3: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y))) ).

% inf_sup_ord(3)
tff(fact_5151_inf__sup__ord_I3_J,axiom,
    ! [X3: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y))) ).

% inf_sup_ord(3)
tff(fact_5152_inf__sup__ord_I3_J,axiom,
    ! [X3: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y))) ).

% inf_sup_ord(3)
tff(fact_5153_le__supE,axiom,
    ! [A: assn,B: assn,X3: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B),X3)
     => ~ ( ord_less_eq_assn(A,X3)
         => ~ ord_less_eq_assn(B,X3) ) ) ).

% le_supE
tff(fact_5154_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X3))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),X3))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),X3)) ) ) ).

% le_supE
tff(fact_5155_le__supE,axiom,
    ! [A: set_int,B: set_int,X3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)),X3))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(A),X3))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(B),X3)) ) ) ).

% le_supE
tff(fact_5156_le__supE,axiom,
    ! [A: rat,B: rat,X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X3))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X3))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),X3)) ) ) ).

% le_supE
tff(fact_5157_le__supE,axiom,
    ! [A: nat,B: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X3))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X3))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),X3)) ) ) ).

% le_supE
tff(fact_5158_le__supE,axiom,
    ! [A: int,B: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X3))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X3))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),X3)) ) ) ).

% le_supE
tff(fact_5159_le__supI,axiom,
    ! [A: assn,X3: assn,B: assn] :
      ( ord_less_eq_assn(A,X3)
     => ( ord_less_eq_assn(B,X3)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B),X3) ) ) ).

% le_supI
tff(fact_5160_le__supI,axiom,
    ! [A: set_nat,X3: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),X3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),X3))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X3)) ) ) ).

% le_supI
tff(fact_5161_le__supI,axiom,
    ! [A: set_int,X3: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),X3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),X3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)),X3)) ) ) ).

% le_supI
tff(fact_5162_le__supI,axiom,
    ! [A: rat,X3: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),X3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X3)) ) ) ).

% le_supI
tff(fact_5163_le__supI,axiom,
    ! [A: nat,X3: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),X3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X3)) ) ) ).

% le_supI
tff(fact_5164_le__supI,axiom,
    ! [A: int,X3: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),X3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X3)) ) ) ).

% le_supI
tff(fact_5165_sup__ge1,axiom,
    ! [X3: assn,Y: assn] : ord_less_eq_assn(X3,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y)) ).

% sup_ge1
tff(fact_5166_sup__ge1,axiom,
    ! [X3: set_nat,Y: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y))) ).

% sup_ge1
tff(fact_5167_sup__ge1,axiom,
    ! [X3: set_int,Y: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y))) ).

% sup_ge1
tff(fact_5168_sup__ge1,axiom,
    ! [X3: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y))) ).

% sup_ge1
tff(fact_5169_sup__ge1,axiom,
    ! [X3: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y))) ).

% sup_ge1
tff(fact_5170_sup__ge1,axiom,
    ! [X3: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y))) ).

% sup_ge1
tff(fact_5171_sup__ge2,axiom,
    ! [Y: assn,X3: assn] : ord_less_eq_assn(Y,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y)) ).

% sup_ge2
tff(fact_5172_sup__ge2,axiom,
    ! [Y: set_nat,X3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y))) ).

% sup_ge2
tff(fact_5173_sup__ge2,axiom,
    ! [Y: set_int,X3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(Y),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y))) ).

% sup_ge2
tff(fact_5174_sup__ge2,axiom,
    ! [Y: rat,X3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y))) ).

% sup_ge2
tff(fact_5175_sup__ge2,axiom,
    ! [Y: nat,X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y))) ).

% sup_ge2
tff(fact_5176_sup__ge2,axiom,
    ! [Y: int,X3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y))) ).

% sup_ge2
tff(fact_5177_le__supI1,axiom,
    ! [X3: assn,A: assn,B: assn] :
      ( ord_less_eq_assn(X3,A)
     => ord_less_eq_assn(X3,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% le_supI1
tff(fact_5178_le__supI1,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),A))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% le_supI1
tff(fact_5179_le__supI1,axiom,
    ! [X3: set_int,A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),A))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B))) ) ).

% le_supI1
tff(fact_5180_le__supI1,axiom,
    ! [X3: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% le_supI1
tff(fact_5181_le__supI1,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% le_supI1
tff(fact_5182_le__supI1,axiom,
    ! [X3: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% le_supI1
tff(fact_5183_le__supI2,axiom,
    ! [X3: assn,B: assn,A: assn] :
      ( ord_less_eq_assn(X3,B)
     => ord_less_eq_assn(X3,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% le_supI2
tff(fact_5184_le__supI2,axiom,
    ! [X3: set_nat,B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),B))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% le_supI2
tff(fact_5185_le__supI2,axiom,
    ! [X3: set_int,B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B))) ) ).

% le_supI2
tff(fact_5186_le__supI2,axiom,
    ! [X3: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% le_supI2
tff(fact_5187_le__supI2,axiom,
    ! [X3: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% le_supI2
tff(fact_5188_le__supI2,axiom,
    ! [X3: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% le_supI2
tff(fact_5189_sup_Omono,axiom,
    ! [C: assn,A: assn,D: assn,B: assn] :
      ( ord_less_eq_assn(C,A)
     => ( ord_less_eq_assn(D,B)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).

% sup.mono
tff(fact_5190_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,D),B))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ) ).

% sup.mono
tff(fact_5191_sup_Omono,axiom,
    ! [C: set_int,A: set_int,D: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(D),B))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C),D)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B))) ) ) ).

% sup.mono
tff(fact_5192_sup_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ) ).

% sup.mono
tff(fact_5193_sup_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ) ).

% sup.mono
tff(fact_5194_sup_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ) ).

% sup.mono
tff(fact_5195_sup__mono,axiom,
    ! [A: assn,C: assn,B: assn,D: assn] :
      ( ord_less_eq_assn(A,C)
     => ( ord_less_eq_assn(B,D)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D)) ) ) ).

% sup_mono
tff(fact_5196_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),C))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),D))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D))) ) ) ).

% sup_mono
tff(fact_5197_sup__mono,axiom,
    ! [A: set_int,C: set_int,B: set_int,D: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),C))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(B),D))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C),D))) ) ) ).

% sup_mono
tff(fact_5198_sup__mono,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D))) ) ) ).

% sup_mono
tff(fact_5199_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D))) ) ) ).

% sup_mono
tff(fact_5200_sup__mono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D))) ) ) ).

% sup_mono
tff(fact_5201_sup__least,axiom,
    ! [Y: assn,X3: assn,Z: assn] :
      ( ord_less_eq_assn(Y,X3)
     => ( ord_less_eq_assn(Z,X3)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z),X3) ) ) ).

% sup_least
tff(fact_5202_sup__least,axiom,
    ! [Y: set_nat,X3: set_nat,Z: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z),X3))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X3)) ) ) ).

% sup_least
tff(fact_5203_sup__least,axiom,
    ! [Y: set_int,X3: set_int,Z: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X3))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(Z),X3))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z)),X3)) ) ) ).

% sup_least
tff(fact_5204_sup__least,axiom,
    ! [Y: rat,X3: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z)),X3)) ) ) ).

% sup_least
tff(fact_5205_sup__least,axiom,
    ! [Y: nat,X3: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X3)) ) ) ).

% sup_least
tff(fact_5206_sup__least,axiom,
    ! [Y: int,X3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),X3)) ) ) ).

% sup_least
tff(fact_5207_le__iff__sup,axiom,
    ! [X3: assn,Y: assn] :
      ( ord_less_eq_assn(X3,Y)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_5208_le__iff__sup,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Y))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_5209_le__iff__sup,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_5210_le__iff__sup,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_5211_le__iff__sup,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_5212_le__iff__sup,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y) = Y ) ) ).

% le_iff_sup
tff(fact_5213_sup_OorderE,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
     => ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).

% sup.orderE
tff(fact_5214_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A))
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.orderE
tff(fact_5215_sup_OorderE,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) ) ) ).

% sup.orderE
tff(fact_5216_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.orderE
tff(fact_5217_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.orderE
tff(fact_5218_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.orderE
tff(fact_5219_sup_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
     => ord_less_eq_assn(B,A) ) ).

% sup.orderI
tff(fact_5220_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A)) ) ).

% sup.orderI
tff(fact_5221_sup_OorderI,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(B),A)) ) ).

% sup.orderI
tff(fact_5222_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% sup.orderI
tff(fact_5223_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% sup.orderI
tff(fact_5224_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% sup.orderI
tff(fact_5225_sup__unique,axiom,
    ! [F: fun_as3983628580203722482n_assn,X3: assn,Y: assn] :
      ( ! [X: assn,Y3: assn] : ord_less_eq_assn(X,aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y3))
     => ( ! [X: assn,Y3: assn] : ord_less_eq_assn(Y3,aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y3))
       => ( ! [X: assn,Y3: assn,Z3: assn] :
              ( ord_less_eq_assn(Y3,X)
             => ( ord_less_eq_assn(Z3,X)
               => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(F,Y3),Z3),X) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_5226_sup__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X3: set_nat,Y: set_nat] :
      ( ! [X: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y3)))
     => ( ! [X: set_nat,Y3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y3)))
       => ( ! [X: set_nat,Y3: set_nat,Z3: set_nat] :
              ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y3),X))
             => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z3),X))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y3),Z3)),X)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_5227_sup__unique,axiom,
    ! [F: fun_se6573205548028396576et_int,X3: set_int,Y: set_int] :
      ( ! [X: set_int,Y3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(F,X),Y3)))
     => ( ! [X: set_int,Y3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(Y3),aa_set_int_set_int(aa_set4234024024286713313et_int(F,X),Y3)))
       => ( ! [X: set_int,Y3: set_int,Z3: set_int] :
              ( pp(aa_set_int_bool(ord_less_eq_set_int(Y3),X))
             => ( pp(aa_set_int_bool(ord_less_eq_set_int(Z3),X))
               => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(F,Y3),Z3)),X)) ) )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y) = aa_set_int_set_int(aa_set4234024024286713313et_int(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_5228_sup__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X3: rat,Y: rat] :
      ( ! [X: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y3)))
     => ( ! [X: rat,Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y3)))
       => ( ! [X: rat,Y3: rat,Z3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X))
             => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z3),X))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(F,Y3),Z3)),X)) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_5229_sup__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X3: nat,Y: nat] :
      ( ! [X: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y3)))
     => ( ! [X: nat,Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y3)))
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z3),X))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y3),Z3)),X)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_5230_sup__unique,axiom,
    ! [F: fun_int_fun_int_int,X3: int,Y: int] :
      ( ! [X: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(F,X),Y3)))
     => ( ! [X: int,Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),aa_int_int(aa_int_fun_int_int(F,X),Y3)))
       => ( ! [X: int,Y3: int,Z3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z3),X))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y3),Z3)),X)) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y) = aa_int_int(aa_int_fun_int_int(F,X3),Y) ) ) ) ) ).

% sup_unique
tff(fact_5231_sup_Oabsorb1,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb1
tff(fact_5232_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_5233_sup_Oabsorb1,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_5234_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_5235_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_5236_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_5237_sup_Oabsorb2,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb2
tff(fact_5238_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_5239_sup_Oabsorb2,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_5240_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_5241_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_5242_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_5243_sup__absorb1,axiom,
    ! [Y: assn,X3: assn] :
      ( ord_less_eq_assn(Y,X3)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_5244_sup__absorb1,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y),X3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_5245_sup__absorb1,axiom,
    ! [Y: set_int,X3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_5246_sup__absorb1,axiom,
    ! [Y: rat,X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_5247_sup__absorb1,axiom,
    ! [Y: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_5248_sup__absorb1,axiom,
    ! [Y: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y) = X3 ) ) ).

% sup_absorb1
tff(fact_5249_sup__absorb2,axiom,
    ! [X3: assn,Y: assn] :
      ( ord_less_eq_assn(X3,Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_5250_sup__absorb2,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Y))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_5251_sup__absorb2,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_5252_sup__absorb2,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_5253_sup__absorb2,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_5254_sup__absorb2,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y) = Y ) ) ).

% sup_absorb2
tff(fact_5255_sup_OboundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C),A)
     => ~ ( ord_less_eq_assn(B,A)
         => ~ ord_less_eq_assn(C,A) ) ) ).

% sup.boundedE
tff(fact_5256_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A)) ) ) ).

% sup.boundedE
tff(fact_5257_sup_OboundedE,axiom,
    ! [B: set_int,C: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)),A))
     => ~ ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
         => ~ pp(aa_set_int_bool(ord_less_eq_set_int(C),A)) ) ) ).

% sup.boundedE
tff(fact_5258_sup_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% sup.boundedE
tff(fact_5259_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% sup.boundedE
tff(fact_5260_sup_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% sup.boundedE
tff(fact_5261_sup_OboundedI,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ord_less_eq_assn(B,A)
     => ( ord_less_eq_assn(C,A)
       => ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C),A) ) ) ).

% sup.boundedI
tff(fact_5262_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)) ) ) ).

% sup.boundedI
tff(fact_5263_sup_OboundedI,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A))
       => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C)),A)) ) ) ).

% sup.boundedI
tff(fact_5264_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)) ) ) ).

% sup.boundedI
tff(fact_5265_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)) ) ) ).

% sup.boundedI
tff(fact_5266_sup_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)) ) ) ).

% sup.boundedI
tff(fact_5267_sup_Oorder__iff,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).

% sup.order_iff
tff(fact_5268_sup_Oorder__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A))
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_5269_sup_Oorder__iff,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
    <=> ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) ) ) ).

% sup.order_iff
tff(fact_5270_sup_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.order_iff
tff(fact_5271_sup_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_5272_sup_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.order_iff
tff(fact_5273_sup_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ord_less_eq_assn(A,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).

% sup.cobounded1
tff(fact_5274_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ).

% sup.cobounded1
tff(fact_5275_sup_Ocobounded1,axiom,
    ! [A: set_int,B: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(A),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B))) ).

% sup.cobounded1
tff(fact_5276_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ).

% sup.cobounded1
tff(fact_5277_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ).

% sup.cobounded1
tff(fact_5278_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ).

% sup.cobounded1
tff(fact_5279_sup_Ocobounded2,axiom,
    ! [B: assn,A: assn] : ord_less_eq_assn(B,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).

% sup.cobounded2
tff(fact_5280_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ).

% sup.cobounded2
tff(fact_5281_sup_Ocobounded2,axiom,
    ! [B: set_int,A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(B),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B))) ).

% sup.cobounded2
tff(fact_5282_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ).

% sup.cobounded2
tff(fact_5283_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ).

% sup.cobounded2
tff(fact_5284_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ).

% sup.cobounded2
tff(fact_5285_sup_Oabsorb__iff1,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_eq_assn(B,A)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_5286_sup_Oabsorb__iff1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B),A))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_5287_sup_Oabsorb__iff1,axiom,
    ! [B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B),A))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_5288_sup_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_5289_sup_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_5290_sup_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_5291_sup_Oabsorb__iff2,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_5292_sup_Oabsorb__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),B))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_5293_sup_Oabsorb__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_5294_sup_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_5295_sup_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_5296_sup_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_5297_sup_OcoboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( ord_less_eq_assn(C,A)
     => ord_less_eq_assn(C,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.coboundedI1
tff(fact_5298_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),A))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% sup.coboundedI1
tff(fact_5299_sup_OcoboundedI1,axiom,
    ! [C: set_int,A: set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C),A))
     => pp(aa_set_int_bool(ord_less_eq_set_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B))) ) ).

% sup.coboundedI1
tff(fact_5300_sup_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% sup.coboundedI1
tff(fact_5301_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% sup.coboundedI1
tff(fact_5302_sup_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% sup.coboundedI1
tff(fact_5303_sup_OcoboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( ord_less_eq_assn(C,B)
     => ord_less_eq_assn(C,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.coboundedI2
tff(fact_5304_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),B))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% sup.coboundedI2
tff(fact_5305_sup_OcoboundedI2,axiom,
    ! [C: set_int,B: set_int,A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(C),B))
     => pp(aa_set_int_bool(ord_less_eq_set_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B))) ) ).

% sup.coboundedI2
tff(fact_5306_sup_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% sup.coboundedI2
tff(fact_5307_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% sup.coboundedI2
tff(fact_5308_sup_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% sup.coboundedI2
tff(fact_5309_sup_Ostrict__coboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( ord_less_assn(C,B)
     => ord_less_assn(C,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_5310_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),B))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% sup.strict_coboundedI2
tff(fact_5311_sup_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(sup_sup_real(A),B))) ) ).

% sup.strict_coboundedI2
tff(fact_5312_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% sup.strict_coboundedI2
tff(fact_5313_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% sup.strict_coboundedI2
tff(fact_5314_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% sup.strict_coboundedI2
tff(fact_5315_sup_Ostrict__coboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( ord_less_assn(C,A)
     => ord_less_assn(C,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_5316_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),A))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% sup.strict_coboundedI1
tff(fact_5317_sup_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(sup_sup_real(A),B))) ) ).

% sup.strict_coboundedI1
tff(fact_5318_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% sup.strict_coboundedI1
tff(fact_5319_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% sup.strict_coboundedI1
tff(fact_5320_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% sup.strict_coboundedI1
tff(fact_5321_sup_Ostrict__order__iff,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_assn(B,A)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5322_sup_Ostrict__order__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B),A))
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5323_sup_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( ( A = aa_real_real(sup_sup_real(A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5324_sup_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5325_sup_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5326_sup_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5327_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C),A)
     => ~ ( ord_less_assn(B,A)
         => ~ ord_less_assn(C,A) ) ) ).

% sup.strict_boundedE
tff(fact_5328_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A))
     => ~ ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B),A))
         => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,C),A)) ) ) ).

% sup.strict_boundedE
tff(fact_5329_sup_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sup_sup_real(B),C)),A))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% sup.strict_boundedE
tff(fact_5330_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% sup.strict_boundedE
tff(fact_5331_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% sup.strict_boundedE
tff(fact_5332_sup_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% sup.strict_boundedE
tff(fact_5333_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_assn(A,B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb4
tff(fact_5334_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,A),B))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_5335_sup_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(sup_sup_real(A),B) = B ) ) ).

% sup.absorb4
tff(fact_5336_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_5337_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_5338_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb4
tff(fact_5339_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( ord_less_assn(B,A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb3
tff(fact_5340_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,B),A))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_5341_sup_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( aa_real_real(sup_sup_real(A),B) = A ) ) ).

% sup.absorb3
tff(fact_5342_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_5343_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_5344_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb3
tff(fact_5345_less__supI2,axiom,
    ! [X3: assn,B: assn,A: assn] :
      ( ord_less_assn(X3,B)
     => ord_less_assn(X3,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI2
tff(fact_5346_less__supI2,axiom,
    ! [X3: set_nat,B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X3),B))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% less_supI2
tff(fact_5347_less__supI2,axiom,
    ! [X3: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(sup_sup_real(A),B))) ) ).

% less_supI2
tff(fact_5348_less__supI2,axiom,
    ! [X3: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% less_supI2
tff(fact_5349_less__supI2,axiom,
    ! [X3: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% less_supI2
tff(fact_5350_less__supI2,axiom,
    ! [X3: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% less_supI2
tff(fact_5351_less__supI1,axiom,
    ! [X3: assn,A: assn,B: assn] :
      ( ord_less_assn(X3,A)
     => ord_less_assn(X3,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI1
tff(fact_5352_less__supI1,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X3),A))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))) ) ).

% less_supI1
tff(fact_5353_less__supI1,axiom,
    ! [X3: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(sup_sup_real(A),B))) ) ).

% less_supI1
tff(fact_5354_less__supI1,axiom,
    ! [X3: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B))) ) ).

% less_supI1
tff(fact_5355_less__supI1,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B))) ) ).

% less_supI1
tff(fact_5356_less__supI1,axiom,
    ! [X3: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B))) ) ).

% less_supI1
tff(fact_5357_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_5358_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1)))
     => ( ! [I2: int,J2: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(I2),J2)))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),J2))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I2),J2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_5359_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q4)),R2) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),L)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R2))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q4 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5360_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila8603258263270017530r_assn(sup_sup_assn,bot_bot_assn,aTP_Lamm_sk,aTP_Lamm_sl) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_5361_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila4459961161675019956t_real(sup_sup_set_real,bot_bot_set_real,aTP_Lamm_sm,aTP_Lamm_sn) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_5362_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila8830438993314920101t_bool(sup_sup_set_bool,bot_bot_set_bool,aTP_Lamm_so,aTP_Lamm_sp) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_5363_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila1667268886620078168et_nat(sup_sup_set_nat,bot_bot_set_nat,aTP_Lamm_ch,aTP_Lamm_sq) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_5364_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila6712789903965657268et_int(sup_sup_set_int,bot_bot_set_int,aTP_Lamm_ck,aTP_Lamm_sr) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_5365_merge__correct,axiom,
    ! [L1: list_real,L22: list_real] :
      ( ( pp(aa_list_real_bool(distinct_real,L1))
        & pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),L1)) )
     => ( ( pp(aa_list_real_bool(distinct_real,L22))
          & pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),L22)) )
       => ( pp(aa_list_real_bool(distinct_real,merge_real(L1,L22)))
          & pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),merge_real(L1,L22)))
          & ( aa_lis3769085826154824284t_real(set_real2,merge_real(L1,L22)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,aa_lis3769085826154824284t_real(set_real2,L1)),aa_lis3769085826154824284t_real(set_real2,L22)) ) ) ) ) ).

% merge_correct
tff(fact_5366_merge__correct,axiom,
    ! [L1: list_bool,L22: list_bool] :
      ( ( pp(aa_list_bool_bool(distinct_bool,L1))
        & pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),L1)) )
     => ( ( pp(aa_list_bool_bool(distinct_bool,L22))
          & pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),L22)) )
       => ( pp(aa_list_bool_bool(distinct_bool,merge_bool(L1,L22)))
          & pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),merge_bool(L1,L22)))
          & ( aa_lis3587183946281582166t_bool(set_bool2,merge_bool(L1,L22)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,aa_lis3587183946281582166t_bool(set_bool2,L1)),aa_lis3587183946281582166t_bool(set_bool2,L22)) ) ) ) ) ).

% merge_correct
tff(fact_5367_merge__correct,axiom,
    ! [L1: list_rat,L22: list_rat] :
      ( ( distinct_rat(L1)
        & sorted_wrt_rat(ord_less_eq_rat,L1) )
     => ( ( distinct_rat(L22)
          & sorted_wrt_rat(ord_less_eq_rat,L22) )
       => ( distinct_rat(merge_rat(L1,L22))
          & sorted_wrt_rat(ord_less_eq_rat,merge_rat(L1,L22))
          & ( set_rat2(merge_rat(L1,L22)) = sup_sup_set_rat(set_rat2(L1),set_rat2(L22)) ) ) ) ) ).

% merge_correct
tff(fact_5368_merge__correct,axiom,
    ! [L1: list_num,L22: list_num] :
      ( ( distinct_num(L1)
        & sorted_wrt_num(ord_less_eq_num,L1) )
     => ( ( distinct_num(L22)
          & sorted_wrt_num(ord_less_eq_num,L22) )
       => ( distinct_num(merge_num(L1,L22))
          & sorted_wrt_num(ord_less_eq_num,merge_num(L1,L22))
          & ( set_num2(merge_num(L1,L22)) = sup_sup_set_num(set_num2(L1),set_num2(L22)) ) ) ) ) ).

% merge_correct
tff(fact_5369_merge__correct,axiom,
    ! [L1: list_nat,L22: list_nat] :
      ( ( pp(aa_list_nat_bool(distinct_nat,L1))
        & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L1)) )
     => ( ( pp(aa_list_nat_bool(distinct_nat,L22))
          & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L22)) )
       => ( pp(aa_list_nat_bool(distinct_nat,merge_nat(L1,L22)))
          & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),merge_nat(L1,L22)))
          & ( aa_list_nat_set_nat(set_nat2,merge_nat(L1,L22)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_list_nat_set_nat(set_nat2,L1)),aa_list_nat_set_nat(set_nat2,L22)) ) ) ) ) ).

% merge_correct
tff(fact_5370_merge__correct,axiom,
    ! [L1: list_int,L22: list_int] :
      ( ( pp(aa_list_int_bool(distinct_int,L1))
        & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L1)) )
     => ( ( pp(aa_list_int_bool(distinct_int,L22))
          & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L22)) )
       => ( pp(aa_list_int_bool(distinct_int,merge_int(L1,L22)))
          & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),merge_int(L1,L22)))
          & ( aa_list_int_set_int(set_int2,merge_int(L1,L22)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_list_int_set_int(set_int2,L1)),aa_list_int_set_int(set_int2,L22)) ) ) ) ) ).

% merge_correct
tff(fact_5371_time__array__nth,axiom,
    ! [P3: array_VEBT_VEBTi,I: nat,H: heap_e7401611519738050253t_unit] :
      ( ( time_f8309030937046825446_VEBTi(array_nth_VEBT_VEBTi(P3,I),H)
       => ( time_time_VEBT_VEBTi(array_nth_VEBT_VEBTi(P3,I),H) = zero_zero_nat ) )
      & ( ~ time_f8309030937046825446_VEBTi(array_nth_VEBT_VEBTi(P3,I),H)
       => ( time_time_VEBT_VEBTi(array_nth_VEBT_VEBTi(P3,I),H) = one_one_nat ) ) ) ).

% time_array_nth
tff(fact_5372_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_5373_abs__idempotent,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_idempotent
tff(fact_5374_abs__idempotent,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).

% abs_idempotent
tff(fact_5375_abs__idempotent,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_idempotent
tff(fact_5376_abs__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_abs
tff(fact_5377_abs__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_abs
tff(fact_5378_abs__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).

% abs_abs
tff(fact_5379_abs__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_abs
tff(fact_5380_abs__0,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_5381_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_5382_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_5383_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_5384_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_5385_abs__zero,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_5386_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_5387_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_5388_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_5389_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_5390_abs__eq__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_5391_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_5392_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_5393_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_5394_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_5395_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_5396_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_5397_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ).

% abs_add_abs
tff(fact_5398_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_5399_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_5400_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_5401_abs__1,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_5402_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_5403_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_5404_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_5405_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A) ).

% abs_mult_self_eq
tff(fact_5406_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% abs_mult_self_eq
tff(fact_5407_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% abs_mult_self_eq
tff(fact_5408_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% abs_mult_self_eq
tff(fact_5409_abs__of__nat,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% abs_of_nat
tff(fact_5410_abs__of__nat,axiom,
    ! [N: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_5411_abs__of__nat,axiom,
    ! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% abs_of_nat
tff(fact_5412_abs__of__nat,axiom,
    ! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% abs_of_nat
tff(fact_5413_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger))
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_5414_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_5415_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_5416_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_5417_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% abs_le_self_iff
tff(fact_5418_abs__le__self__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% abs_le_self_iff
tff(fact_5419_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% abs_le_self_iff
tff(fact_5420_abs__le__self__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% abs_le_self_iff
tff(fact_5421_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5422_abs__of__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5423_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5424_abs__of__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5425_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_5426_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A)))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_5427_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_5428_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A)))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_5429_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int))
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_5430_norm__assertion__simps_I15_J,axiom,
    ! [X3: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% norm_assertion_simps(15)
tff(fact_5431_norm__assertion__simps_I32_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),X3) = X3 ).

% norm_assertion_simps(32)
tff(fact_5432_assn__aci_I5_J,axiom,
    ! [X3: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X3) ).

% assn_aci(5)
tff(fact_5433_assn__aci_I7_J,axiom,
    ! [X3: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Z)) ).

% assn_aci(7)
tff(fact_5434_assn__aci_I8_J,axiom,
    ! [X3: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y) ).

% assn_aci(8)
tff(fact_5435_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_5436_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_5437_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_5438_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_5439_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_5440_abs__ge__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(abs_abs_real,A))) ).

% abs_ge_self
tff(fact_5441_abs__ge__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))) ).

% abs_ge_self
tff(fact_5442_abs__ge__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_self
tff(fact_5443_abs__ge__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(abs_abs_int,A))) ).

% abs_ge_self
tff(fact_5444_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% abs_le_D1
tff(fact_5445_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% abs_le_D1
tff(fact_5446_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% abs_le_D1
tff(fact_5447_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% abs_le_D1
tff(fact_5448_abs__one,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_5449_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_5450_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_5451_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_5452_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ).

% abs_mult
tff(fact_5453_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_5454_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_5455_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_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,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_5456_abs__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_mult
tff(fact_5457_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)) ).

% abs_minus_commute
tff(fact_5458_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)) ).

% abs_minus_commute
tff(fact_5459_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)) ).

% abs_minus_commute
tff(fact_5460_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_5461_eucl__rel__int__by0,axiom,
    ! [K: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K))) ).

% eucl_rel_int_by0
tff(fact_5462_abs__ge__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))) ).

% abs_ge_zero
tff(fact_5463_abs__ge__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A))) ).

% abs_ge_zero
tff(fact_5464_abs__ge__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_zero
tff(fact_5465_abs__ge__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A))) ).

% abs_ge_zero
tff(fact_5466_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),zero_z3403309356797280102nteger)) ).

% abs_not_less_zero
tff(fact_5467_abs__not__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real)) ).

% abs_not_less_zero
tff(fact_5468_abs__not__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)) ).

% abs_not_less_zero
tff(fact_5469_abs__not__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int)) ).

% abs_not_less_zero
tff(fact_5470_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ).

% abs_of_pos
tff(fact_5471_abs__of__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_5472_abs__of__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_5473_abs__of__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_5474_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)))) ).

% abs_triangle_ineq
tff(fact_5475_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq
tff(fact_5476_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq
tff(fact_5477_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq
tff(fact_5478_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),D))) ) ) ).

% abs_mult_less
tff(fact_5479_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,B)),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))) ) ) ).

% abs_mult_less
tff(fact_5480_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D))) ) ) ).

% abs_mult_less
tff(fact_5481_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,B)),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))) ) ) ).

% abs_mult_less
tff(fact_5482_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5483_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5484_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5485_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5486_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)))) ).

% abs_triangle_ineq3
tff(fact_5487_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)))) ).

% abs_triangle_ineq3
tff(fact_5488_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)))) ).

% abs_triangle_ineq3
tff(fact_5489_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq3
tff(fact_5490_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B)))) ).

% abs_triangle_ineq2
tff(fact_5491_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)))) ).

% abs_triangle_ineq2
tff(fact_5492_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)))) ).

% abs_triangle_ineq2
tff(fact_5493_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq2
tff(fact_5494_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
    <=> ! [M5: int] :
        ? [N6: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M5),aa_int_int(abs_abs_int,N6)))
          & pp(aa_set_int_bool(member_int(N6),S3)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_5495_dense__eq0__I,axiom,
    ! [X3: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),E)) )
     => ( X3 = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_5496_dense__eq0__I,axiom,
    ! [X3: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X3)),E)) )
     => ( X3 = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5497_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ) ) ).

% abs_eq_mult
tff(fact_5498_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_5499_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_5500_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
     => ( aa_int_int(abs_abs_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,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_5501_abs__mult__pos,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3))
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y)),X3) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),X3)) ) ) ).

% abs_mult_pos
tff(fact_5502_abs__mult__pos,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),X3) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X3)) ) ) ).

% abs_mult_pos
tff(fact_5503_abs__mult__pos,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),X3) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X3)) ) ) ).

% abs_mult_pos
tff(fact_5504_abs__mult__pos,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X3) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X3)) ) ) ).

% abs_mult_pos
tff(fact_5505_merge_Osimps_I3_J,axiom,
    ! [X1: real,X2: real,L1: list_real,L22: list_real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X1),X2))
       => ( merge_real(cons_real(X1,L1),cons_real(X2,L22)) = cons_real(X1,merge_real(L1,cons_real(X2,L22))) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X1),X2))
       => ( ( ( X1 = X2 )
           => ( merge_real(cons_real(X1,L1),cons_real(X2,L22)) = cons_real(X1,merge_real(L1,L22)) ) )
          & ( ( X1 != X2 )
           => ( merge_real(cons_real(X1,L1),cons_real(X2,L22)) = cons_real(X2,merge_real(cons_real(X1,L1),L22)) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_5506_merge_Osimps_I3_J,axiom,
    ! [X1: rat,X2: rat,L1: list_rat,L22: list_rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X1),X2))
       => ( merge_rat(cons_rat(X1,L1),cons_rat(X2,L22)) = cons_rat(X1,merge_rat(L1,cons_rat(X2,L22))) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X1),X2))
       => ( ( ( X1 = X2 )
           => ( merge_rat(cons_rat(X1,L1),cons_rat(X2,L22)) = cons_rat(X1,merge_rat(L1,L22)) ) )
          & ( ( X1 != X2 )
           => ( merge_rat(cons_rat(X1,L1),cons_rat(X2,L22)) = cons_rat(X2,merge_rat(cons_rat(X1,L1),L22)) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_5507_merge_Osimps_I3_J,axiom,
    ! [X1: num,X2: num,L1: list_num,L22: list_num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X1),X2))
       => ( merge_num(cons_num(X1,L1),cons_num(X2,L22)) = cons_num(X1,merge_num(L1,cons_num(X2,L22))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X1),X2))
       => ( ( ( X1 = X2 )
           => ( merge_num(cons_num(X1,L1),cons_num(X2,L22)) = cons_num(X1,merge_num(L1,L22)) ) )
          & ( ( X1 != X2 )
           => ( merge_num(cons_num(X1,L1),cons_num(X2,L22)) = cons_num(X2,merge_num(cons_num(X1,L1),L22)) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_5508_merge_Osimps_I3_J,axiom,
    ! [X1: nat,X2: nat,L1: list_nat,L22: list_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X1),X2))
       => ( merge_nat(aa_list_nat_list_nat(cons_nat(X1),L1),aa_list_nat_list_nat(cons_nat(X2),L22)) = aa_list_nat_list_nat(cons_nat(X1),merge_nat(L1,aa_list_nat_list_nat(cons_nat(X2),L22))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X1),X2))
       => ( ( ( X1 = X2 )
           => ( merge_nat(aa_list_nat_list_nat(cons_nat(X1),L1),aa_list_nat_list_nat(cons_nat(X2),L22)) = aa_list_nat_list_nat(cons_nat(X1),merge_nat(L1,L22)) ) )
          & ( ( X1 != X2 )
           => ( merge_nat(aa_list_nat_list_nat(cons_nat(X1),L1),aa_list_nat_list_nat(cons_nat(X2),L22)) = aa_list_nat_list_nat(cons_nat(X2),merge_nat(aa_list_nat_list_nat(cons_nat(X1),L1),L22)) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_5509_merge_Osimps_I3_J,axiom,
    ! [X1: int,X2: int,L1: list_int,L22: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X1),X2))
       => ( merge_int(aa_list_int_list_int(cons_int(X1),L1),aa_list_int_list_int(cons_int(X2),L22)) = aa_list_int_list_int(cons_int(X1),merge_int(L1,aa_list_int_list_int(cons_int(X2),L22))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X1),X2))
       => ( ( ( X1 = X2 )
           => ( merge_int(aa_list_int_list_int(cons_int(X1),L1),aa_list_int_list_int(cons_int(X2),L22)) = aa_list_int_list_int(cons_int(X1),merge_int(L1,L22)) ) )
          & ( ( X1 != X2 )
           => ( merge_int(aa_list_int_list_int(cons_int(X1),L1),aa_list_int_list_int(cons_int(X2),L22)) = aa_list_int_list_int(cons_int(X2),merge_int(aa_list_int_list_int(cons_int(X1),L1),L22)) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_5510_abs__diff__le__iff,axiom,
    ! [X3: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),R2)),X3))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X3),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5511_abs__diff__le__iff,axiom,
    ! [X3: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X3),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(A),R2)),X3))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5512_abs__diff__le__iff,axiom,
    ! [X3: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R2)),X3))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5513_abs__diff__le__iff,axiom,
    ! [X3: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X3))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5514_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),B))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)))) ).

% abs_triangle_ineq4
tff(fact_5515_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq4
tff(fact_5516_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq4
tff(fact_5517_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq4
tff(fact_5518_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(C),D)))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),C))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_5519_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_5520_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_5521_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_5522_abs__diff__less__iff,axiom,
    ! [X3: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X3),A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A),R2)),X3))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5523_abs__diff__less__iff,axiom,
    ! [X3: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X3),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(minus_minus_real(A),R2)),X3))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5524_abs__diff__less__iff,axiom,
    ! [X3: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R2)),X3))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5525_abs__diff__less__iff,axiom,
    ! [X3: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X3))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5526_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L) )
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),zero_zero_int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5527_abs__add__one__gt__zero,axiom,
    ! [X3: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X3)))) ).

% abs_add_one_gt_zero
tff(fact_5528_abs__add__one__gt__zero,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,X3)))) ).

% abs_add_one_gt_zero
tff(fact_5529_abs__add__one__gt__zero,axiom,
    ! [X3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X3)))) ).

% abs_add_one_gt_zero
tff(fact_5530_abs__add__one__gt__zero,axiom,
    ! [X3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X3)))) ).

% abs_add_one_gt_zero
tff(fact_5531_time__refines,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,C3: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit] :
      ( refine5565527176597971370_VEBTi(C,C3)
     => ( ~ time_f8309030937046825446_VEBTi(C3,H)
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(C,H)),time_time_VEBT_VEBTi(C3,H))) ) ) ).

% time_refines
tff(fact_5532_time__refines,axiom,
    ! [C: heap_Time_Heap_bool,C3: heap_Time_Heap_bool,H: heap_e7401611519738050253t_unit] :
      ( refine4616567020171210338s_bool(C,C3)
     => ( ~ time_fails_bool(C3,H)
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_bool(C,H)),time_time_bool(C3,H))) ) ) ).

% time_refines
tff(fact_5533_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
           => ? [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_5534_decr__lemma,axiom,
    ! [D: int,X3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),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,X3),Z))),one_one_int)),D))),Z)) ) ).

% decr_lemma
tff(fact_5535_incr__lemma,axiom,
    ! [D: int,Z: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),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,X3),Z))),one_one_int)),D)))) ) ).

% incr_lemma
tff(fact_5536_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_5537_bezw__0,axiom,
    ! [X3: nat] : bezw(X3,zero_zero_nat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_5538_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K3) ) )
        | ? [L3: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L3) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L3)))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_5539_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22)))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_5540_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( linord2324613341767563021nteger(minus_2355218937544613996nteger(A2,insert_Code_integer(X3,bot_bo3990330152332043303nteger))) = remove1_Code_integer(X3,linord2324613341767563021nteger(A2)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5541_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( linord4252657396651189596t_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) = remove1_real(X3,linord4252657396651189596t_real(A2)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5542_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( linord193515670363874663t_bool(minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) = remove1_bool(X3,linord193515670363874663t_bool(A2)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5543_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( linord2612477271533052124et_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) = remove1_int(X3,linord2612477271533052124et_int(A2)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5544_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) = remove1_nat(X3,aa_set_nat_list_nat(linord2614967742042102400et_nat,A2)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5545_sgn__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(sgn_sgn_int,A)) = aa_int_int(sgn_sgn_int,A) ).

% sgn_sgn
tff(fact_5546_sgn__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% sgn_sgn
tff(fact_5547_sgn__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ).

% sgn_sgn
tff(fact_5548_sgn__sgn,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(sgn_sgn_rat,A)) = aa_rat_rat(sgn_sgn_rat,A) ).

% sgn_sgn
tff(fact_5549_sgn__0,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_5550_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_5551_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_5552_sgn__0,axiom,
    aa_rat_rat(sgn_sgn_rat,zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_5553_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_5554_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_5555_sgn__1,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_1
tff(fact_5556_sgn__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_5557_sgn__1,axiom,
    aa_rat_rat(sgn_sgn_rat,one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_5558_sgn__greater,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% sgn_greater
tff(fact_5559_sgn__greater,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% sgn_greater
tff(fact_5560_sgn__greater,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% sgn_greater
tff(fact_5561_sgn__greater,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% sgn_greater
tff(fact_5562_sgn__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% sgn_less
tff(fact_5563_sgn__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% sgn_less
tff(fact_5564_sgn__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(sgn_sgn_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% sgn_less
tff(fact_5565_sgn__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% sgn_less
tff(fact_5566_sgn__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_5567_sgn__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) ) ).

% sgn_pos
tff(fact_5568_sgn__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_5569_sgn__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_5570_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_5571_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_5572_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_5573_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_5574_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_5575_sgn__0__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_5576_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_5577_sgn__0__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_5578_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_5579_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_5580_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_5581_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_5582_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_5583_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ) ) ).

% same_sgn_sgn_add
tff(fact_5584_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(sgn_sgn_real,A) ) ) ).

% same_sgn_sgn_add
tff(fact_5585_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_rat(sgn_sgn_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(sgn_sgn_rat,A) ) ) ).

% same_sgn_sgn_add
tff(fact_5586_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_5587_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ).

% sgn_mult
tff(fact_5588_sgn__mult,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_mult
tff(fact_5589_sgn__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,B)) ).

% sgn_mult
tff(fact_5590_sgn__mult,axiom,
    ! [A: int,B: int] : aa_int_int(sgn_sgn_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,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).

% sgn_mult
tff(fact_5591_sgn__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_mult
tff(fact_5592_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A),B)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)) ) ) ).

% same_sgn_abs_add
tff(fact_5593_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% same_sgn_abs_add
tff(fact_5594_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% same_sgn_abs_add
tff(fact_5595_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% same_sgn_abs_add
tff(fact_5596_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_5597_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: real] : aa_real_real(abs_abs_real,K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),aa_real_real(sgn_sgn_real,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_5598_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: rat] : aa_rat_rat(abs_abs_rat,K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),aa_rat_rat(sgn_sgn_rat,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_5599_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_5600_abs__mult__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = A ).

% abs_mult_sgn
tff(fact_5601_abs__mult__sgn,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(sgn_sgn_real,A)) = A ).

% abs_mult_sgn
tff(fact_5602_abs__mult__sgn,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = A ).

% abs_mult_sgn
tff(fact_5603_abs__mult__sgn,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).

% abs_mult_sgn
tff(fact_5604_abs__mult__sgn,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(sgn_sgn_complex,A)) = A ).

% abs_mult_sgn
tff(fact_5605_sgn__mult__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)) = A ).

% sgn_mult_abs
tff(fact_5606_sgn__mult__abs,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(abs_abs_real,A)) = A ).

% sgn_mult_abs
tff(fact_5607_sgn__mult__abs,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(abs_abs_rat,A)) = A ).

% sgn_mult_abs
tff(fact_5608_sgn__mult__abs,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).

% sgn_mult_abs
tff(fact_5609_sgn__mult__abs,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(abs_abs_complex,A)) = A ).

% sgn_mult_abs
tff(fact_5610_mult__sgn__abs,axiom,
    ! [X3: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X3)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X3)) = X3 ).

% mult_sgn_abs
tff(fact_5611_mult__sgn__abs,axiom,
    ! [X3: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X3)),aa_real_real(abs_abs_real,X3)) = X3 ).

% mult_sgn_abs
tff(fact_5612_mult__sgn__abs,axiom,
    ! [X3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,X3)),aa_rat_rat(abs_abs_rat,X3)) = X3 ).

% mult_sgn_abs
tff(fact_5613_mult__sgn__abs,axiom,
    ! [X3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,X3)),aa_int_int(abs_abs_int,X3)) = X3 ).

% mult_sgn_abs
tff(fact_5614_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A2: set_rat] : sorted_wrt_rat(ord_less_eq_rat,linord1979837681955606664et_rat(A2)) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5615_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A2: set_num] : sorted_wrt_num(ord_less_eq_num,linord8395671565052656842et_num(A2)) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5616_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A2: set_int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),linord2612477271533052124et_int(A2))) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5617_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A2: set_nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),aa_set_nat_list_nat(linord2614967742042102400et_nat,A2))) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5618_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A2: set_real] : pp(aa_list_real_bool(sorted_wrt_real(ord_less_real),linord4252657396651189596t_real(A2))) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5619_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A2: set_rat] : sorted_wrt_rat(ord_less_rat,linord1979837681955606664et_rat(A2)) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5620_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A2: set_num] : sorted_wrt_num(ord_less_num,linord8395671565052656842et_num(A2)) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5621_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A2: set_int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),linord2612477271533052124et_int(A2))) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5622_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A2: set_nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),aa_set_nat_list_nat(linord2614967742042102400et_nat,A2))) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5623_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% sgn_1_pos
tff(fact_5624_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = one_one_real )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% sgn_1_pos
tff(fact_5625_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat )
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% sgn_1_pos
tff(fact_5626_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% sgn_1_pos
tff(fact_5627_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      ( ( ( A = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_z3403309356797280102nteger ) )
      & ( ( A != zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
tff(fact_5628_abs__sgn__eq,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ) ).

% abs_sgn_eq
tff(fact_5629_abs__sgn__eq,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ) ).

% abs_sgn_eq
tff(fact_5630_abs__sgn__eq,axiom,
    ! [A: int] :
      ( ( ( A = zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = zero_zero_int ) )
      & ( ( A != zero_zero_int )
       => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ) ).

% abs_sgn_eq
tff(fact_5631_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs))
     => ( pp(aa_list_real_bool(distinct_real,Xs))
       => ( linord4252657396651189596t_real(aa_lis3769085826154824284t_real(set_real2,Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5632_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
     => ( pp(aa_list_bool_bool(distinct_bool,Xs))
       => ( linord193515670363874663t_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5633_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_5634_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_5635_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => ( pp(aa_list_int_bool(distinct_int,Xs))
       => ( linord2612477271533052124et_int(aa_list_int_set_int(set_int2,Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5636_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => ( pp(aa_list_nat_bool(distinct_nat,Xs))
       => ( aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_list_nat_set_nat(set_nat2,Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5637_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_5638_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_5639_lemma__interval,axiom,
    ! [A: real,X3: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X3),Y4))),D2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),B)) ) ) ) ) ) ).

% lemma_interval
tff(fact_5640_sin__bound__lemma,axiom,
    ! [X3: real,Y: real,U: real,V: real] :
      ( ( X3 = Y )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),U)),Y))),V)) ) ) ).

% sin_bound_lemma
tff(fact_5641_lemma__interval__lt,axiom,
    ! [A: real,X3: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
       => ? [D2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X3),Y4))),D2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),B)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_5642_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [X3: real,Y: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X3)),aa_real_real(sgn_sgn_real,Y)) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_5643_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [X3: complex,Y: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sgn_sgn_complex,X3)),aa_complex_complex(sgn_sgn_complex,Y)) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_5644_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( linord2324613341767563021nteger(insert_Code_integer(X3,A2)) = linord7786967269120229815nteger(aTP_Lamm_ss,X3,linord2324613341767563021nteger(minus_2355218937544613996nteger(A2,insert_Code_integer(X3,bot_bo3990330152332043303nteger)))) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5645_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( linord4252657396651189596t_real(aa_set_real_set_real(insert_real(X3),A2)) = linord1674302359176591317l_real(aTP_Lamm_dj,X3,linord4252657396651189596t_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5646_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( linord193515670363874663t_bool(aa_set_bool_set_bool(insert_bool(X3),A2)) = linord1990632362053644219l_bool(aTP_Lamm_st,X3,linord193515670363874663t_bool(minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)))) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5647_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( linord2612477271533052124et_int(aa_set_int_set_int(insert_int(X3),A2)) = linord734827384618529109nt_int(aTP_Lamm_dm,X3,linord2612477271533052124et_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)))) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5648_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_set_nat_set_nat(insert_nat(X3),A2)) = linord8961336180081300637at_nat(aTP_Lamm_dl,X3,aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)))) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5649_sgn__zero__iff,axiom,
    ! [X3: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,X3) = zero_zero_complex )
    <=> ( X3 = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_5650_sgn__zero__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(sgn_sgn_real,X3) = zero_zero_real )
    <=> ( X3 = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_5651_sgn__le__0__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sgn_sgn_real,X3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real)) ) ).

% sgn_le_0_iff
tff(fact_5652_zero__le__sgn__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3)) ) ).

% zero_le_sgn_iff
tff(fact_5653_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ~ pp(aa_set_bool_bool(member_bool(X3),A2))
       => ( linord193515670363874663t_bool(aa_set_bool_set_bool(insert_bool(X3),A2)) = linord1990632362053644219l_bool(aTP_Lamm_st,X3,linord193515670363874663t_bool(A2)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_5654_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X3),A2))
       => ( linord4252657396651189596t_real(aa_set_real_set_real(insert_real(X3),A2)) = linord1674302359176591317l_real(aTP_Lamm_dj,X3,linord4252657396651189596t_real(A2)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_5655_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X3),A2))
       => ( linord2612477271533052124et_int(aa_set_int_set_int(insert_int(X3),A2)) = linord734827384618529109nt_int(aTP_Lamm_dm,X3,linord2612477271533052124et_int(A2)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_5656_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A2))
       => ( linord2324613341767563021nteger(insert_Code_integer(X3,A2)) = linord7786967269120229815nteger(aTP_Lamm_ss,X3,linord2324613341767563021nteger(A2)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_5657_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X3),A2))
       => ( aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_set_nat_set_nat(insert_nat(X3),A2)) = linord8961336180081300637at_nat(aTP_Lamm_dl,X3,aa_set_nat_list_nat(linord2614967742042102400et_nat,A2)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_5658_insort__key_Osimps_I2_J,axiom,
    ! [F: fun_nat_rat,X3: nat,Y: nat,Ys: list_nat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)))
       => ( linord8326206119994804901at_rat(F,X3,aa_list_nat_list_nat(cons_nat(Y),Ys)) = aa_list_nat_list_nat(cons_nat(X3),aa_list_nat_list_nat(cons_nat(Y),Ys)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)))
       => ( linord8326206119994804901at_rat(F,X3,aa_list_nat_list_nat(cons_nat(Y),Ys)) = aa_list_nat_list_nat(cons_nat(Y),linord8326206119994804901at_rat(F,X3,Ys)) ) ) ) ).

% insort_key.simps(2)
tff(fact_5659_insort__key_Osimps_I2_J,axiom,
    ! [F: fun_int_rat,X3: int,Y: int,Ys: list_int] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)))
       => ( linord102187795041083649nt_rat(F,X3,aa_list_int_list_int(cons_int(Y),Ys)) = aa_list_int_list_int(cons_int(X3),aa_list_int_list_int(cons_int(Y),Ys)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)))
       => ( linord102187795041083649nt_rat(F,X3,aa_list_int_list_int(cons_int(Y),Ys)) = aa_list_int_list_int(cons_int(Y),linord102187795041083649nt_rat(F,X3,Ys)) ) ) ) ).

% insort_key.simps(2)
tff(fact_5660_insort__key_Osimps_I2_J,axiom,
    ! [F: fun_nat_num,X3: nat,Y: nat,Ys: list_nat] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)))
       => ( linord5518667966237079271at_num(F,X3,aa_list_nat_list_nat(cons_nat(Y),Ys)) = aa_list_nat_list_nat(cons_nat(X3),aa_list_nat_list_nat(cons_nat(Y),Ys)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)))
       => ( linord5518667966237079271at_num(F,X3,aa_list_nat_list_nat(cons_nat(Y),Ys)) = aa_list_nat_list_nat(cons_nat(Y),linord5518667966237079271at_num(F,X3,Ys)) ) ) ) ).

% insort_key.simps(2)
tff(fact_5661_insort__key_Osimps_I2_J,axiom,
    ! [F: fun_int_num,X3: int,Y: int,Ys: list_int] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X3)),aa_int_num(F,Y)))
       => ( linord6518021678138133827nt_num(F,X3,aa_list_int_list_int(cons_int(Y),Ys)) = aa_list_int_list_int(cons_int(X3),aa_list_int_list_int(cons_int(Y),Ys)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X3)),aa_int_num(F,Y)))
       => ( linord6518021678138133827nt_num(F,X3,aa_list_int_list_int(cons_int(Y),Ys)) = aa_list_int_list_int(cons_int(Y),linord6518021678138133827nt_num(F,X3,Ys)) ) ) ) ).

% insort_key.simps(2)
tff(fact_5662_insort__key_Osimps_I2_J,axiom,
    ! [F: fun_nat_nat,X3: nat,Y: nat,Ys: list_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y)))
       => ( linord8961336180081300637at_nat(F,X3,aa_list_nat_list_nat(cons_nat(Y),Ys)) = aa_list_nat_list_nat(cons_nat(X3),aa_list_nat_list_nat(cons_nat(Y),Ys)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y)))
       => ( linord8961336180081300637at_nat(F,X3,aa_list_nat_list_nat(cons_nat(Y),Ys)) = aa_list_nat_list_nat(cons_nat(Y),linord8961336180081300637at_nat(F,X3,Ys)) ) ) ) ).

% insort_key.simps(2)
tff(fact_5663_insort__key_Osimps_I2_J,axiom,
    ! [F: fun_int_nat,X3: int,Y: int,Ys: list_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(F,Y)))
       => ( linord737317855127579385nt_nat(F,X3,aa_list_int_list_int(cons_int(Y),Ys)) = aa_list_int_list_int(cons_int(X3),aa_list_int_list_int(cons_int(Y),Ys)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(F,Y)))
       => ( linord737317855127579385nt_nat(F,X3,aa_list_int_list_int(cons_int(Y),Ys)) = aa_list_int_list_int(cons_int(Y),linord737317855127579385nt_nat(F,X3,Ys)) ) ) ) ).

% insort_key.simps(2)
tff(fact_5664_insort__key_Osimps_I2_J,axiom,
    ! [F: fun_nat_int,X3: nat,Y: nat,Ys: list_nat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(F,Y)))
       => ( linord8958845709572250361at_int(F,X3,aa_list_nat_list_nat(cons_nat(Y),Ys)) = aa_list_nat_list_nat(cons_nat(X3),aa_list_nat_list_nat(cons_nat(Y),Ys)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(F,Y)))
       => ( linord8958845709572250361at_int(F,X3,aa_list_nat_list_nat(cons_nat(Y),Ys)) = aa_list_nat_list_nat(cons_nat(Y),linord8958845709572250361at_int(F,X3,Ys)) ) ) ) ).

% insort_key.simps(2)
tff(fact_5665_insort__key_Osimps_I2_J,axiom,
    ! [F: fun_int_int,X3: int,Y: int,Ys: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(F,Y)))
       => ( linord734827384618529109nt_int(F,X3,aa_list_int_list_int(cons_int(Y),Ys)) = aa_list_int_list_int(cons_int(X3),aa_list_int_list_int(cons_int(Y),Ys)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(F,Y)))
       => ( linord734827384618529109nt_int(F,X3,aa_list_int_list_int(cons_int(Y),Ys)) = aa_list_int_list_int(cons_int(Y),linord734827384618529109nt_int(F,X3,Ys)) ) ) ) ).

% insort_key.simps(2)
tff(fact_5666_sorted__insort,axiom,
    ! [X3: rat,Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,linord8009651476059938989at_rat(aTP_Lamm_dk,X3,Xs))
    <=> sorted_wrt_rat(ord_less_eq_rat,Xs) ) ).

% sorted_insort
tff(fact_5667_sorted__insort,axiom,
    ! [X3: num,Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,linord3648998369544548657um_num(aTP_Lamm_su,X3,Xs))
    <=> sorted_wrt_num(ord_less_eq_num,Xs) ) ).

% sorted_insort
tff(fact_5668_sorted__insort,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),linord8961336180081300637at_nat(aTP_Lamm_dl,X3,Xs)))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs)) ) ).

% sorted_insort
tff(fact_5669_sorted__insort,axiom,
    ! [X3: int,Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),linord734827384618529109nt_int(aTP_Lamm_dm,X3,Xs)))
    <=> pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs)) ) ).

% sorted_insort
tff(fact_5670_insort__is__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_rat,A: vEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,A)),aa_VEBT_VEBT_rat(F,X))) )
     => ( linord2597477761626359633BT_rat(F,A,Xs) = cons_VEBT_VEBT(A,Xs) ) ) ).

% insort_is_Cons
tff(fact_5671_insort__is__Cons,axiom,
    ! [Xs: list_real,F: fun_real_rat,A: real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,A)),aa_real_rat(F,X))) )
     => ( linord5462632957955297153al_rat(F,A,Xs) = cons_real(A,Xs) ) ) ).

% insort_is_Cons
tff(fact_5672_insort__is__Cons,axiom,
    ! [Xs: list_bool,F: fun_bool_rat,A: bool] :
      ( ! [X: bool] :
          ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_bool_rat(F,A)),aa_bool_rat(F,X))) )
     => ( linord4598724016734318900ol_rat(F,A,Xs) = cons_bool(A,Xs) ) ) ).

% insort_is_Cons
tff(fact_5673_insort__is__Cons,axiom,
    ! [Xs: list_nat,F: fun_nat_rat,A: nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),aa_nat_rat(F,X))) )
     => ( linord8326206119994804901at_rat(F,A,Xs) = aa_list_nat_list_nat(cons_nat(A),Xs) ) ) ).

% insort_is_Cons
tff(fact_5674_insort__is__Cons,axiom,
    ! [Xs: list_int,F: fun_int_rat,A: int] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,A)),aa_int_rat(F,X))) )
     => ( linord102187795041083649nt_rat(F,A,Xs) = aa_list_int_list_int(cons_int(A),Xs) ) ) ).

% insort_is_Cons
tff(fact_5675_insort__is__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_num,A: vEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_VEBT_VEBT_num(F,A)),aa_VEBT_VEBT_num(F,X))) )
     => ( linord9013311644723409811BT_num(F,A,Xs) = cons_VEBT_VEBT(A,Xs) ) ) ).

% insort_is_Cons
tff(fact_5676_insort__is__Cons,axiom,
    ! [Xs: list_real,F: fun_real_num,A: real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,A)),aa_real_num(F,X))) )
     => ( linord2655094804197571523al_num(F,A,Xs) = cons_real(A,Xs) ) ) ).

% insort_is_Cons
tff(fact_5677_insort__is__Cons,axiom,
    ! [Xs: list_bool,F: fun_bool_num,A: bool] :
      ( ! [X: bool] :
          ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_bool_num(F,A)),aa_bool_num(F,X))) )
     => ( linord1791185862976593270ol_num(F,A,Xs) = cons_bool(A,Xs) ) ) ).

% insort_is_Cons
tff(fact_5678_insort__is__Cons,axiom,
    ! [Xs: list_nat,F: fun_nat_num,A: nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),aa_nat_num(F,X))) )
     => ( linord5518667966237079271at_num(F,A,Xs) = aa_list_nat_list_nat(cons_nat(A),Xs) ) ) ).

% insort_is_Cons
tff(fact_5679_insort__is__Cons,axiom,
    ! [Xs: list_int,F: fun_int_num,A: int] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,A)),aa_int_num(F,X))) )
     => ( linord6518021678138133827nt_num(F,A,Xs) = aa_list_int_list_int(cons_int(A),Xs) ) ) ).

% insort_is_Cons
tff(fact_5680_sorted__insort__key,axiom,
    ! [F: fun_VEBT_VEBT_real,X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),map_VEBT_VEBT_real(F,linord210457350419892133T_real(F,X3,Xs))))
    <=> pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),map_VEBT_VEBT_real(F,Xs))) ) ).

% sorted_insort_key
tff(fact_5681_sorted__insort__key,axiom,
    ! [F: fun_Pr4298194452408323251l_bool,X3: produc2804947525362035943l_bool,Xs: list_P2722043638750504429l_bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),map_Pr8574941368599290932l_bool(F,linord3512919210420388587l_bool(F,X3,Xs))))
    <=> pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),map_Pr8574941368599290932l_bool(F,Xs))) ) ).

% sorted_insort_key
tff(fact_5682_sorted__insort__key,axiom,
    ! [F: fun_nat_bool,X3: nat,Xs: list_nat] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),map_nat_bool(F,linord273055089713377930t_bool(F,X3,Xs))))
    <=> pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),map_nat_bool(F,Xs))) ) ).

% sorted_insort_key
tff(fact_5683_sorted__insort__key,axiom,
    ! [F: fun_VEBT_VEBT_nat,X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_VEBT_VEBT_nat(F,linord3232607821712855369BT_nat(F,X3,Xs))))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_VEBT_VEBT_nat(F,Xs))) ) ).

% sorted_insort_key
tff(fact_5684_sorted__insort__key,axiom,
    ! [F: fun_nat_nat,X3: nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_nat_nat(F,linord8961336180081300637at_nat(F,X3,Xs))))
    <=> pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),map_nat_nat(F,Xs))) ) ).

% sorted_insort_key
tff(fact_5685_mult__commute__abs,axiom,
    ! [C: real] : aTP_Lamm_sv(C) = aa_rea7340605660300375871l_real(times_times_real,C) ).

% mult_commute_abs
tff(fact_5686_mult__commute__abs,axiom,
    ! [C: rat] : aa_rat_fun_rat_rat(aTP_Lamm_sw,C) = aa_rat_fun_rat_rat(times_times_rat,C) ).

% mult_commute_abs
tff(fact_5687_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_sx,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_5688_mult__commute__abs,axiom,
    ! [C: int] : aa_int_fun_int_int(aTP_Lamm_sy,C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_5689_mult__commute__abs,axiom,
    ! [C: assn] : aa_ass954072293579886259n_assn(aTP_Lamm_sz,C) = aa_ass954072293579886259n_assn(times_times_assn,C) ).

% mult_commute_abs
tff(fact_5690_mult__commute__abs,axiom,
    ! [C: complex] : aa_com3342990543769683525omplex(aTP_Lamm_ta,C) = aa_com3342990543769683525omplex(times_times_complex,C) ).

% mult_commute_abs
tff(fact_5691_insort__remove1,axiom,
    ! [A: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(A),aa_lis3769085826154824284t_real(set_real2,Xs)))
     => ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs))
       => ( linord1674302359176591317l_real(aTP_Lamm_dj,A,remove1_real(A,Xs)) = Xs ) ) ) ).

% insort_remove1
tff(fact_5692_insort__remove1,axiom,
    ! [A: bool,Xs: list_bool] :
      ( pp(aa_set_bool_bool(member_bool(A),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
     => ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
       => ( linord1990632362053644219l_bool(aTP_Lamm_st,A,remove1_bool(A,Xs)) = Xs ) ) ) ).

% insort_remove1
tff(fact_5693_insort__remove1,axiom,
    ! [A: rat,Xs: list_rat] :
      ( pp(aa_set_rat_bool(member_rat(A),set_rat2(Xs)))
     => ( sorted_wrt_rat(ord_less_eq_rat,Xs)
       => ( linord8009651476059938989at_rat(aTP_Lamm_dk,A,remove1_rat(A,Xs)) = Xs ) ) ) ).

% insort_remove1
tff(fact_5694_insort__remove1,axiom,
    ! [A: num,Xs: list_num] :
      ( pp(aa_set_num_bool(member_num(A),set_num2(Xs)))
     => ( sorted_wrt_num(ord_less_eq_num,Xs)
       => ( linord3648998369544548657um_num(aTP_Lamm_su,A,remove1_num(A,Xs)) = Xs ) ) ) ).

% insort_remove1
tff(fact_5695_insort__remove1,axiom,
    ! [A: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),aa_list_nat_set_nat(set_nat2,Xs)))
     => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
       => ( linord8961336180081300637at_nat(aTP_Lamm_dl,A,remove1_nat(A,Xs)) = Xs ) ) ) ).

% insort_remove1
tff(fact_5696_insort__remove1,axiom,
    ! [A: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(A),aa_list_int_set_int(set_int2,Xs)))
     => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
       => ( linord734827384618529109nt_int(aTP_Lamm_dm,A,remove1_int(A,Xs)) = Xs ) ) ) ).

% insort_remove1
tff(fact_5697_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A2))
       => ( linord2324613341767563021nteger(A2) = linord7786967269120229815nteger(aTP_Lamm_ss,X3,linord2324613341767563021nteger(minus_2355218937544613996nteger(A2,insert_Code_integer(X3,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5698_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X3),A2))
       => ( linord4252657396651189596t_real(A2) = linord1674302359176591317l_real(aTP_Lamm_dj,X3,linord4252657396651189596t_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5699_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( pp(aa_set_bool_bool(member_bool(X3),A2))
       => ( linord193515670363874663t_bool(A2) = linord1990632362053644219l_bool(aTP_Lamm_st,X3,linord193515670363874663t_bool(minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)))) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5700_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X3),A2))
       => ( linord2612477271533052124et_int(A2) = linord734827384618529109nt_int(aTP_Lamm_dm,X3,linord2612477271533052124et_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)))) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5701_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(X3),A2))
       => ( aa_set_nat_list_nat(linord2614967742042102400et_nat,A2) = linord8961336180081300637at_nat(aTP_Lamm_dl,X3,aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)))) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5702_sorted__list__of__set__def,axiom,
    linord2614967742042102400et_nat = linord1089935798310486446at_nat(aTP_Lamm_dl) ).

% sorted_list_of_set_def
tff(fact_5703_fun__upd__None__restrict,axiom,
    ! [X3: vEBT_VEBT,D4: set_VEBT_VEBT,M: fun_VE8174548813682830439on_nat] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),D4))
       => ( fun_up5885881570350532375on_nat(restri774867724463461460BT_nat(M,D4),X3,none_nat) = restri774867724463461460BT_nat(M,minus_5127226145743854075T_VEBT(D4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),D4))
       => ( fun_up5885881570350532375on_nat(restri774867724463461460BT_nat(M,D4),X3,none_nat) = restri774867724463461460BT_nat(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5704_fun__upd__None__restrict,axiom,
    ! [X3: vEBT_VEBT,D4: set_VEBT_VEBT,M: fun_VE3889768403122177201on_num] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),D4))
       => ( fun_up6594125507299370081on_num(restri6555571547474015902BT_num(M,D4),X3,none_num) = restri6555571547474015902BT_num(M,minus_5127226145743854075T_VEBT(D4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),D4))
       => ( fun_up6594125507299370081on_num(restri6555571547474015902BT_num(M,D4),X3,none_num) = restri6555571547474015902BT_num(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5705_fun__upd__None__restrict,axiom,
    ! [X3: real,D4: set_real,M: fun_real_option_nat] :
      ( ( pp(aa_set_real_bool(member_real(X3),D4))
       => ( fun_up6677080212936659659on_nat(restri6827137924477938990al_nat(M,D4),X3,none_nat) = restri6827137924477938990al_nat(M,minus_minus_set_real(D4,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),D4))
       => ( fun_up6677080212936659659on_nat(restri6827137924477938990al_nat(M,D4),X3,none_nat) = restri6827137924477938990al_nat(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5706_fun__upd__None__restrict,axiom,
    ! [X3: real,D4: set_real,M: fun_real_option_num] :
      ( ( pp(aa_set_real_bool(member_real(X3),D4))
       => ( fun_up7385324149885497365on_num(restri3384469710633717624al_num(M,D4),X3,none_num) = restri3384469710633717624al_num(M,minus_minus_set_real(D4,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),D4))
       => ( fun_up7385324149885497365on_num(restri3384469710633717624al_num(M,D4),X3,none_num) = restri3384469710633717624al_num(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5707_fun__upd__None__restrict,axiom,
    ! [X3: bool,D4: set_bool,M: fun_bool_option_nat] :
      ( ( pp(aa_set_bool_bool(member_bool(X3),D4))
       => ( fun_up4506005684929255546on_nat(restri5405821841207159351ol_nat(M,D4),X3,none_nat) = restri5405821841207159351ol_nat(M,minus_minus_set_bool(D4,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(X3),D4))
       => ( fun_up4506005684929255546on_nat(restri5405821841207159351ol_nat(M,D4),X3,none_nat) = restri5405821841207159351ol_nat(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5708_fun__upd__None__restrict,axiom,
    ! [X3: bool,D4: set_bool,M: fun_bool_option_num] :
      ( ( pp(aa_set_bool_bool(member_bool(X3),D4))
       => ( fun_up5214249621878093252on_num(restri1963153627362937985ol_num(M,D4),X3,none_num) = restri1963153627362937985ol_num(M,minus_minus_set_bool(D4,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(X3),D4))
       => ( fun_up5214249621878093252on_num(restri1963153627362937985ol_num(M,D4),X3,none_num) = restri1963153627362937985ol_num(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5709_fun__upd__None__restrict,axiom,
    ! [X3: int,D4: set_int,M: fun_int_option_nat] :
      ( ( pp(aa_set_int_bool(member_int(X3),D4))
       => ( fun_up3620524117960394059on_nat(restrict_map_int_nat(M,D4),X3,none_nat) = restrict_map_int_nat(M,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,D4),aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X3),D4))
       => ( fun_up3620524117960394059on_nat(restrict_map_int_nat(M,D4),X3,none_nat) = restrict_map_int_nat(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5710_fun__upd__None__restrict,axiom,
    ! [X3: int,D4: set_int,M: fun_int_option_num] :
      ( ( pp(aa_set_int_bool(member_int(X3),D4))
       => ( fun_up4328768054909231765on_num(restrict_map_int_num(M,D4),X3,none_num) = restrict_map_int_num(M,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,D4),aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X3),D4))
       => ( fun_up4328768054909231765on_num(restrict_map_int_num(M,D4),X3,none_num) = restrict_map_int_num(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5711_fun__upd__None__restrict,axiom,
    ! [X3: nat,D4: set_nat,M: fun_nat_option_nat] :
      ( ( pp(aa_set_nat_bool(member_nat(X3),D4))
       => ( fun_up1493157387958331631on_nat(restrict_map_nat_nat(M,D4),X3,none_nat) = restrict_map_nat_nat(M,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,D4),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X3),D4))
       => ( fun_up1493157387958331631on_nat(restrict_map_nat_nat(M,D4),X3,none_nat) = restrict_map_nat_nat(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5712_fun__upd__None__restrict,axiom,
    ! [X3: nat,D4: set_nat,M: fun_nat_option_num] :
      ( ( pp(aa_set_nat_bool(member_nat(X3),D4))
       => ( fun_up2201401324907169337on_num(restrict_map_nat_num(M,D4),X3,none_num) = restrict_map_nat_num(M,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,D4),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X3),D4))
       => ( fun_up2201401324907169337on_num(restrict_map_nat_num(M,D4),X3,none_num) = restrict_map_nat_num(M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5713_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tb,C)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,X3),Y)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
           => ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tb,C)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,X3),Y)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y))
           => ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tb,C)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,X3),Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_5714_total__on__singleton,axiom,
    ! [X3: vEBT_VEBT] : total_on_VEBT_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT),insert494605675473494903T_VEBT(aa_VEB896440568906691887T_VEBT(produc537772716801021591T_VEBT(X3),X3),bot_bo5088076668136028147T_VEBT)) ).

% total_on_singleton
tff(fact_5715_total__on__singleton,axiom,
    ! [X3: code_integer] : total_8516208958838685657nteger(insert_Code_integer(X3,bot_bo3990330152332043303nteger),aa_set6745602309116264572nteger(insert4913895101485356395nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X3),X3)),bot_bo4276436098303576167nteger)) ).

% total_on_singleton
tff(fact_5716_total__on__singleton,axiom,
    ! [X3: real] : total_on_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real),insert7746734233410687241l_real(aa_rea5330357063016736847l_real(produc4511245868158468465l_real(X3),X3),bot_bo3948376660626123781l_real)) ).

% total_on_singleton
tff(fact_5717_total__on__singleton,axiom,
    ! [X3: bool] : total_on_bool(aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool),insert3967033758906133175l_bool(aa_boo3115168258980966476l_bool(produc8297744024737662807l_bool(X3),X3),bot_bo3663083580490196787l_bool)) ).

% total_on_singleton
tff(fact_5718_total__on__singleton,axiom,
    ! [X3: nat] : total_on_nat(aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3)),bot_bo2099793752762293965at_nat)) ).

% total_on_singleton
tff(fact_5719_total__on__singleton,axiom,
    ! [X3: int] : total_on_int(aa_set_int_set_int(insert_int(X3),bot_bot_set_int),aa_set6728540191533482748nt_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),X3)),bot_bo1796632182523588997nt_int)) ).

% total_on_singleton
tff(fact_5720_nth__step__trancl,axiom,
    ! [Xs: list_VEBT_VEBTi,R: set_Pr2227491710730465451_VEBTi,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),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,N2)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N2)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
         => member660371905731732212_VEBTi(produc436343169921013763_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),M)),transi2803566869205510612_VEBTi(R)) ) ) ) ).

% nth_step_trancl
tff(fact_5721_nth__step__trancl,axiom,
    ! [Xs: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat)))
         => member568628332442017744T_VEBT(aa_VEB896440568906691887T_VEBT(produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(suc,N2))),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N2)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
         => member568628332442017744T_VEBT(aa_VEB896440568906691887T_VEBT(produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),M)),transi8906537157094044885T_VEBT(R)) ) ) ) ).

% nth_step_trancl
tff(fact_5722_nth__step__trancl,axiom,
    ! [Xs: list_Code_integer,R: set_Pr4811707699266497531nteger,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s3445333598471063425nteger(Xs)),one_one_nat)))
         => member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(suc,N2))),aa_nat_Code_integer(nth_Code_integer(Xs),N2)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3445333598471063425nteger(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
         => member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),N)),aa_nat_Code_integer(nth_Code_integer(Xs),M)),transi6870300401645067644nteger(R)) ) ) ) ).

% nth_step_trancl
tff(fact_5723_nth__step__trancl,axiom,
    ! [Xs: list_real,R: set_Pr6218003697084177305l_real,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_real_nat(size_size_list_real,Xs)),one_one_nat)))
         => member7849222048561428706l_real(aa_rea5330357063016736847l_real(produc4511245868158468465l_real(aa_nat_real(nth_real(Xs),aa_nat_nat(suc,N2))),aa_nat_real(nth_real(Xs),N2)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
         => member7849222048561428706l_real(aa_rea5330357063016736847l_real(produc4511245868158468465l_real(aa_nat_real(nth_real(Xs),N)),aa_nat_real(nth_real(Xs),M)),transi1789104906590519371l_real(R)) ) ) ) ).

% nth_step_trancl
tff(fact_5724_nth__step__trancl,axiom,
    ! [Xs: list_bool,R: set_Pr8621012493645324487l_bool,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_bool_nat(size_size_list_bool,Xs)),one_one_nat)))
         => member4869097156692242192l_bool(aa_boo3115168258980966476l_bool(produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),aa_nat_nat(suc,N2))),aa_nat_bool(nth_bool(Xs),N2)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
         => member4869097156692242192l_bool(aa_boo3115168258980966476l_bool(produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),N)),aa_nat_bool(nth_bool(Xs),M)),transi8112060559632522936l_bool(R)) ) ) ) ).

% nth_step_trancl
tff(fact_5725_nth__step__trancl,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),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,N2))),aa_nat_nat(nth_nat(Xs),N2)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),N)),aa_nat_nat(nth_nat(Xs),M)),transi6264000038957366511cl_nat(R)) ) ) ) ).

% nth_step_trancl
tff(fact_5726_nth__step__trancl,axiom,
    ! [Xs: list_int,R: set_Pr958786334691620121nt_int,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_int_nat(size_size_list_int,Xs)),one_one_nat)))
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(suc,N2))),aa_nat_int(nth_int(Xs),N2)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),N)),aa_nat_int(nth_int(Xs),M)),transi6261509568448316235cl_int(R)) ) ) ) ).

% nth_step_trancl
tff(fact_5727_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_5728_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_5729_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Ys)))
       => ( nth_Pr316670251186196177_VEBTi(zip_VE6444338338598820466_VEBTi(Xs,Ys),I) = produc6084888613844515218_VEBTi(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5730_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ys)))
       => ( nth_Pr4953567300277697838T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),I) = aa_VEB896440568906691887T_VEBT(produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5731_nth__zip,axiom,
    ! [I: nat,Xs: list_Code_integer,Ys: list_Code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3445333598471063425nteger(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3445333598471063425nteger(Ys)))
       => ( nth_Pr2304437835452373666nteger(zip_Co3543743374963494515nteger(Xs,Ys),I) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),I)),aa_nat_Code_integer(nth_Code_integer(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5732_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(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_5733_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Ys)))
       => ( nth_Pr6842391030413306568T_real(zip_VEBT_VEBT_real(Xs,Ys),I) = aa_rea4189226818901770745T_real(produc8117437818029410057T_real(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_real(nth_real(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5734_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Ys)))
       => ( nth_Pr1878200693239305296i_bool(zip_VEBT_VEBTi_bool(Xs,Ys),I) = produc7282046070707253577i_bool(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I),aa_nat_bool(nth_bool(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5735_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Ys)))
       => ( nth_Pr1289831921885633937T_bool(zip_VEBT_VEBT_bool(Xs,Ys),I) = aa_boo2420671634680568175T_bool(produc3035007008238225530T_bool(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_bool(nth_bool(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5736_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ys)))
       => ( nth_Pr6911489093701683181Ti_nat(zip_VEBT_VEBTi_nat(Xs,Ys),I) = produc7192665754729510430Ti_nat(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I),aa_nat_nat(nth_nat(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5737_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_5738_prod__encode__eq,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(nat_prod_encode,X3) = aa_Pro4243173770044397973at_nat(nat_prod_encode,Y) )
    <=> ( X3 = Y ) ) ).

% prod_encode_eq
tff(fact_5739_image__ident,axiom,
    ! [Y6: set_nat] : aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_dl),Y6) = Y6 ).

% image_ident
tff(fact_5740_image__ident,axiom,
    ! [Y6: set_int] : aa_set_int_set_int(image_int_int(aTP_Lamm_dm),Y6) = Y6 ).

% image_ident
tff(fact_5741_image__is__empty,axiom,
    ! [F: fun_real_real,A2: set_real] :
      ( ( image_real_real(F,A2) = bot_bot_set_real )
    <=> ( A2 = bot_bot_set_real ) ) ).

% image_is_empty
tff(fact_5742_image__is__empty,axiom,
    ! [F: fun_bool_real,A2: set_bool] :
      ( ( image_bool_real(F,A2) = bot_bot_set_real )
    <=> ( A2 = bot_bot_set_bool ) ) ).

% image_is_empty
tff(fact_5743_image__is__empty,axiom,
    ! [F: fun_nat_real,A2: set_nat] :
      ( ( aa_set_nat_set_real(image_nat_real(F),A2) = bot_bot_set_real )
    <=> ( A2 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_5744_image__is__empty,axiom,
    ! [F: fun_int_real,A2: set_int] :
      ( ( image_int_real(F,A2) = bot_bot_set_real )
    <=> ( A2 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_5745_image__is__empty,axiom,
    ! [F: fun_real_bool,A2: set_real] :
      ( ( image_real_bool(F,A2) = bot_bot_set_bool )
    <=> ( A2 = bot_bot_set_real ) ) ).

% image_is_empty
tff(fact_5746_image__is__empty,axiom,
    ! [F: fun_bool_bool,A2: set_bool] :
      ( ( image_bool_bool(F,A2) = bot_bot_set_bool )
    <=> ( A2 = bot_bot_set_bool ) ) ).

% image_is_empty
tff(fact_5747_image__is__empty,axiom,
    ! [F: fun_nat_bool,A2: set_nat] :
      ( ( image_nat_bool(F,A2) = bot_bot_set_bool )
    <=> ( A2 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_5748_image__is__empty,axiom,
    ! [F: fun_int_bool,A2: set_int] :
      ( ( image_int_bool(F,A2) = bot_bot_set_bool )
    <=> ( A2 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_5749_image__is__empty,axiom,
    ! [F: fun_real_nat,A2: set_real] :
      ( ( image_real_nat(F,A2) = bot_bot_set_nat )
    <=> ( A2 = bot_bot_set_real ) ) ).

% image_is_empty
tff(fact_5750_image__is__empty,axiom,
    ! [F: fun_bool_nat,A2: set_bool] :
      ( ( image_bool_nat(F,A2) = bot_bot_set_nat )
    <=> ( A2 = bot_bot_set_bool ) ) ).

% image_is_empty
tff(fact_5751_empty__is__image,axiom,
    ! [F: fun_real_real,A2: set_real] :
      ( ( bot_bot_set_real = image_real_real(F,A2) )
    <=> ( A2 = bot_bot_set_real ) ) ).

% empty_is_image
tff(fact_5752_empty__is__image,axiom,
    ! [F: fun_bool_real,A2: set_bool] :
      ( ( bot_bot_set_real = image_bool_real(F,A2) )
    <=> ( A2 = bot_bot_set_bool ) ) ).

% empty_is_image
tff(fact_5753_empty__is__image,axiom,
    ! [F: fun_nat_real,A2: set_nat] :
      ( ( bot_bot_set_real = aa_set_nat_set_real(image_nat_real(F),A2) )
    <=> ( A2 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_5754_empty__is__image,axiom,
    ! [F: fun_int_real,A2: set_int] :
      ( ( bot_bot_set_real = image_int_real(F,A2) )
    <=> ( A2 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_5755_empty__is__image,axiom,
    ! [F: fun_real_bool,A2: set_real] :
      ( ( bot_bot_set_bool = image_real_bool(F,A2) )
    <=> ( A2 = bot_bot_set_real ) ) ).

% empty_is_image
tff(fact_5756_empty__is__image,axiom,
    ! [F: fun_bool_bool,A2: set_bool] :
      ( ( bot_bot_set_bool = image_bool_bool(F,A2) )
    <=> ( A2 = bot_bot_set_bool ) ) ).

% empty_is_image
tff(fact_5757_empty__is__image,axiom,
    ! [F: fun_nat_bool,A2: set_nat] :
      ( ( bot_bot_set_bool = image_nat_bool(F,A2) )
    <=> ( A2 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_5758_empty__is__image,axiom,
    ! [F: fun_int_bool,A2: set_int] :
      ( ( bot_bot_set_bool = image_int_bool(F,A2) )
    <=> ( A2 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_5759_empty__is__image,axiom,
    ! [F: fun_real_nat,A2: set_real] :
      ( ( bot_bot_set_nat = image_real_nat(F,A2) )
    <=> ( A2 = bot_bot_set_real ) ) ).

% empty_is_image
tff(fact_5760_empty__is__image,axiom,
    ! [F: fun_bool_nat,A2: set_bool] :
      ( ( bot_bot_set_nat = image_bool_nat(F,A2) )
    <=> ( A2 = bot_bot_set_bool ) ) ).

% empty_is_image
tff(fact_5761_image__empty,axiom,
    ! [F: fun_real_real] : image_real_real(F,bot_bot_set_real) = bot_bot_set_real ).

% image_empty
tff(fact_5762_image__empty,axiom,
    ! [F: fun_real_bool] : image_real_bool(F,bot_bot_set_real) = bot_bot_set_bool ).

% image_empty
tff(fact_5763_image__empty,axiom,
    ! [F: fun_real_nat] : image_real_nat(F,bot_bot_set_real) = bot_bot_set_nat ).

% image_empty
tff(fact_5764_image__empty,axiom,
    ! [F: fun_real_int] : image_real_int(F,bot_bot_set_real) = bot_bot_set_int ).

% image_empty
tff(fact_5765_image__empty,axiom,
    ! [F: fun_bool_real] : image_bool_real(F,bot_bot_set_bool) = bot_bot_set_real ).

% image_empty
tff(fact_5766_image__empty,axiom,
    ! [F: fun_bool_bool] : image_bool_bool(F,bot_bot_set_bool) = bot_bot_set_bool ).

% image_empty
tff(fact_5767_image__empty,axiom,
    ! [F: fun_bool_nat] : image_bool_nat(F,bot_bot_set_bool) = bot_bot_set_nat ).

% image_empty
tff(fact_5768_image__empty,axiom,
    ! [F: fun_bool_int] : image_bool_int(F,bot_bot_set_bool) = bot_bot_set_int ).

% image_empty
tff(fact_5769_image__empty,axiom,
    ! [F: fun_nat_real] : aa_set_nat_set_real(image_nat_real(F),bot_bot_set_nat) = bot_bot_set_real ).

% image_empty
tff(fact_5770_image__empty,axiom,
    ! [F: fun_nat_bool] : image_nat_bool(F,bot_bot_set_nat) = bot_bot_set_bool ).

% image_empty
tff(fact_5771_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_real,A5: list_real,B5: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,A) = aa_list_real_nat(size_size_list_real,B) )
     => ( ( aa_list_real_nat(size_size_list_real,A5) = aa_list_real_nat(size_size_list_real,B5) )
       => ( ( zip_real_real(A,B) = zip_real_real(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5772_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_bool,A5: list_real,B5: list_bool] :
      ( ( aa_list_real_nat(size_size_list_real,A) = aa_list_bool_nat(size_size_list_bool,B) )
     => ( ( aa_list_real_nat(size_size_list_real,A5) = aa_list_bool_nat(size_size_list_bool,B5) )
       => ( ( zip_real_bool(A,B) = zip_real_bool(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5773_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_nat,A5: list_real,B5: list_nat] :
      ( ( aa_list_real_nat(size_size_list_real,A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( aa_list_real_nat(size_size_list_real,A5) = aa_list_nat_nat(size_size_list_nat,B5) )
       => ( ( zip_real_nat(A,B) = zip_real_nat(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5774_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_int,A5: list_real,B5: list_int] :
      ( ( aa_list_real_nat(size_size_list_real,A) = aa_list_int_nat(size_size_list_int,B) )
     => ( ( aa_list_real_nat(size_size_list_real,A5) = aa_list_int_nat(size_size_list_int,B5) )
       => ( ( zip_real_int(A,B) = zip_real_int(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5775_zip__eq__zip__same__len,axiom,
    ! [A: list_bool,B: list_real,A5: list_bool,B5: list_real] :
      ( ( aa_list_bool_nat(size_size_list_bool,A) = aa_list_real_nat(size_size_list_real,B) )
     => ( ( aa_list_bool_nat(size_size_list_bool,A5) = aa_list_real_nat(size_size_list_real,B5) )
       => ( ( zip_bool_real(A,B) = zip_bool_real(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5776_zip__eq__zip__same__len,axiom,
    ! [A: list_bool,B: list_bool,A5: list_bool,B5: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,A) = aa_list_bool_nat(size_size_list_bool,B) )
     => ( ( aa_list_bool_nat(size_size_list_bool,A5) = aa_list_bool_nat(size_size_list_bool,B5) )
       => ( ( zip_bool_bool(A,B) = zip_bool_bool(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5777_zip__eq__zip__same__len,axiom,
    ! [A: list_bool,B: list_nat,A5: list_bool,B5: list_nat] :
      ( ( aa_list_bool_nat(size_size_list_bool,A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( aa_list_bool_nat(size_size_list_bool,A5) = aa_list_nat_nat(size_size_list_nat,B5) )
       => ( ( zip_bool_nat(A,B) = zip_bool_nat(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5778_zip__eq__zip__same__len,axiom,
    ! [A: list_bool,B: list_int,A5: list_bool,B5: list_int] :
      ( ( aa_list_bool_nat(size_size_list_bool,A) = aa_list_int_nat(size_size_list_int,B) )
     => ( ( aa_list_bool_nat(size_size_list_bool,A5) = aa_list_int_nat(size_size_list_int,B5) )
       => ( ( zip_bool_int(A,B) = zip_bool_int(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5779_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_real,A5: list_nat,B5: list_real] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = aa_list_real_nat(size_size_list_real,B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A5) = aa_list_real_nat(size_size_list_real,B5) )
       => ( ( zip_nat_real(A,B) = zip_nat_real(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5780_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_bool,A5: list_nat,B5: list_bool] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = aa_list_bool_nat(size_size_list_bool,B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A5) = aa_list_bool_nat(size_size_list_bool,B5) )
       => ( ( zip_nat_bool(A,B) = zip_nat_bool(A5,B5) )
        <=> ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5781_image__add__0,axiom,
    ! [S3: set_complex] : image_1468599708987790691omplex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),S3) = S3 ).

% image_add_0
tff(fact_5782_image__add__0,axiom,
    ! [S3: set_real] : image_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),S3) = S3 ).

% image_add_0
tff(fact_5783_image__add__0,axiom,
    ! [S3: set_rat] : image_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),S3) = S3 ).

% image_add_0
tff(fact_5784_image__add__0,axiom,
    ! [S3: set_nat] : aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat)),S3) = S3 ).

% image_add_0
tff(fact_5785_image__add__0,axiom,
    ! [S3: set_int] : aa_set_int_set_int(image_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int)),S3) = S3 ).

% image_add_0
tff(fact_5786_restrict__map__to__empty,axiom,
    ! [M: fun_real_option_nat,X5: real] : aa_real_option_nat(restri6827137924477938990al_nat(M,bot_bot_set_real),X5) = none_nat ).

% restrict_map_to_empty
tff(fact_5787_restrict__map__to__empty,axiom,
    ! [M: fun_real_option_num,X5: real] : aa_real_option_num(restri3384469710633717624al_num(M,bot_bot_set_real),X5) = none_num ).

% restrict_map_to_empty
tff(fact_5788_restrict__map__to__empty,axiom,
    ! [M: fun_bool_option_nat,X5: bool] : aa_bool_option_nat(restri5405821841207159351ol_nat(M,bot_bot_set_bool),X5) = none_nat ).

% restrict_map_to_empty
tff(fact_5789_restrict__map__to__empty,axiom,
    ! [M: fun_bool_option_num,X5: bool] : aa_bool_option_num(restri1963153627362937985ol_num(M,bot_bot_set_bool),X5) = none_num ).

% restrict_map_to_empty
tff(fact_5790_restrict__map__to__empty,axiom,
    ! [M: fun_nat_option_nat,X5: nat] : aa_nat_option_nat(restrict_map_nat_nat(M,bot_bot_set_nat),X5) = none_nat ).

% restrict_map_to_empty
tff(fact_5791_restrict__map__to__empty,axiom,
    ! [M: fun_nat_option_num,X5: nat] : aa_nat_option_num(restrict_map_nat_num(M,bot_bot_set_nat),X5) = none_num ).

% restrict_map_to_empty
tff(fact_5792_restrict__map__to__empty,axiom,
    ! [M: fun_int_option_nat,X5: int] : aa_int_option_nat(restrict_map_int_nat(M,bot_bot_set_int),X5) = none_nat ).

% restrict_map_to_empty
tff(fact_5793_restrict__map__to__empty,axiom,
    ! [M: fun_int_option_num,X5: int] : aa_int_option_num(restrict_map_int_num(M,bot_bot_set_int),X5) = none_num ).

% restrict_map_to_empty
tff(fact_5794_restrict__map__to__empty,axiom,
    ! [M: fun_re3480452875996755060at_nat,X5: real] : aa_rea4028291186853080669at_nat(restri4688977514882730531at_nat(M,bot_bot_set_real),X5) = none_P5556105721700978146at_nat ).

% restrict_map_to_empty
tff(fact_5795_restrict__map__to__empty,axiom,
    ! [M: fun_bo8911862826410830003at_nat,X5: bool] : aa_boo8810255509353819820at_nat(restri2700956433160322522at_nat(M,bot_bot_set_bool),X5) = none_P5556105721700978146at_nat ).

% restrict_map_to_empty
tff(fact_5796_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I),J)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_5797_image__add__atLeastLessThan_H,axiom,
    ! [K: real,I: real,J: real] : image_real_real(aTP_Lamm_tc(K),aa_real_set_real(set_or66887138388493659n_real(I),J)) = aa_real_set_real(set_or66887138388493659n_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)) ).

% image_add_atLeastLessThan'
tff(fact_5798_image__add__atLeastLessThan_H,axiom,
    ! [K: rat,I: rat,J: rat] : image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_td,K),set_or4029947393144176647an_rat(I,J)) = set_or4029947393144176647an_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K)) ).

% image_add_atLeastLessThan'
tff(fact_5799_image__add__atLeastLessThan_H,axiom,
    ! [K: nat,I: nat,J: nat] : aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_te,K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I),J)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_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)) ).

% image_add_atLeastLessThan'
tff(fact_5800_image__add__atLeastLessThan_H,axiom,
    ! [K: int,I: int,J: int] : aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_tf,K)),aa_int_set_int(set_or4662586982721622107an_int(I),J)) = aa_int_set_int(set_or4662586982721622107an_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),K)) ).

% image_add_atLeastLessThan'
tff(fact_5801_image__add__atLeastLessThan_H,axiom,
    ! [K: code_integer,I: code_integer,J: code_integer] : image_4470545334726330049nteger(aTP_Lamm_tg(K),set_or8404916559141939852nteger(I,J)) = set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(I),K),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(J),K)) ).

% image_add_atLeastLessThan'
tff(fact_5802_trancl__single,axiom,
    ! [A: nat,B: nat] : transi6264000038957366511cl_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),bot_bo2099793752762293965at_nat)) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),bot_bo2099793752762293965at_nat) ).

% trancl_single
tff(fact_5803_trancl__single,axiom,
    ! [A: int,B: int] : transi6261509568448316235cl_int(aa_set6728540191533482748nt_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B)),bot_bo1796632182523588997nt_int)) = aa_set6728540191533482748nt_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A),B)),bot_bo1796632182523588997nt_int) ).

% trancl_single
tff(fact_5804_trancl__single,axiom,
    ! [A: code_integer,B: code_integer] : transi6870300401645067644nteger(aa_set6745602309116264572nteger(insert4913895101485356395nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(A),B)),bot_bo4276436098303576167nteger)) = aa_set6745602309116264572nteger(insert4913895101485356395nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(A),B)),bot_bo4276436098303576167nteger) ).

% trancl_single
tff(fact_5805_nth__image__indices,axiom,
    ! [L: list_VEBT_VEBTi] : image_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(L))) = set_VEBT_VEBTi2(L) ).

% nth_image_indices
tff(fact_5806_nth__image__indices,axiom,
    ! [L: list_set_nat] : aa_set7246812951530217472et_nat(image_nat_set_nat(nth_set_nat(L)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s3254054031482475050et_nat(L))) = set_set_nat2(L) ).

% nth_image_indices
tff(fact_5807_nth__image__indices,axiom,
    ! [L: list_VEBT_VEBT] : image_nat_VEBT_VEBT(nth_VEBT_VEBT(L),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(L))) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L) ).

% nth_image_indices
tff(fact_5808_nth__image__indices,axiom,
    ! [L: list_real] : aa_set_nat_set_real(image_nat_real(nth_real(L)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,L))) = aa_lis3769085826154824284t_real(set_real2,L) ).

% nth_image_indices
tff(fact_5809_nth__image__indices,axiom,
    ! [L: list_bool] : image_nat_bool(nth_bool(L),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_bool_nat(size_size_list_bool,L))) = aa_lis3587183946281582166t_bool(set_bool2,L) ).

% nth_image_indices
tff(fact_5810_nth__image__indices,axiom,
    ! [L: list_nat] : aa_set_nat_set_nat(image_nat_nat(nth_nat(L)),aa_nat_set_nat(aa_nat1818925681583257441et_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_5811_nth__image__indices,axiom,
    ! [L: list_int] : aa_set_nat_set_int(image_nat_int(nth_int(L)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_int_nat(size_size_list_int,L))) = aa_list_int_set_int(set_int2,L) ).

% nth_image_indices
tff(fact_5812_imageE,axiom,
    ! [B: nat,F: fun_nat_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),A2)))
     => ~ ! [X: nat] :
            ( ( B = aa_nat_nat(F,X) )
           => ~ pp(aa_set_nat_bool(member_nat(X),A2)) ) ) ).

% imageE
tff(fact_5813_imageE,axiom,
    ! [B: nat,F: fun_VEBT_VEBT_nat,A2: set_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(B),image_VEBT_VEBT_nat(F,A2)))
     => ~ ! [X: vEBT_VEBT] :
            ( ( B = aa_VEBT_VEBT_nat(F,X) )
           => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2)) ) ) ).

% imageE
tff(fact_5814_imageE,axiom,
    ! [B: nat,F: fun_real_nat,A2: set_real] :
      ( pp(aa_set_nat_bool(member_nat(B),image_real_nat(F,A2)))
     => ~ ! [X: real] :
            ( ( B = aa_real_nat(F,X) )
           => ~ pp(aa_set_real_bool(member_real(X),A2)) ) ) ).

% imageE
tff(fact_5815_imageE,axiom,
    ! [B: nat,F: fun_int_nat,A2: set_int] :
      ( pp(aa_set_nat_bool(member_nat(B),image_int_nat(F,A2)))
     => ~ ! [X: int] :
            ( ( B = aa_int_nat(F,X) )
           => ~ pp(aa_set_int_bool(member_int(X),A2)) ) ) ).

% imageE
tff(fact_5816_imageE,axiom,
    ! [B: vEBT_VEBT,F: fun_nat_VEBT_VEBT,A2: set_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),image_nat_VEBT_VEBT(F,A2)))
     => ~ ! [X: nat] :
            ( ( B = aa_nat_VEBT_VEBT(F,X) )
           => ~ pp(aa_set_nat_bool(member_nat(X),A2)) ) ) ).

% imageE
tff(fact_5817_imageE,axiom,
    ! [B: vEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),image_3375948659692109573T_VEBT(F,A2)))
     => ~ ! [X: vEBT_VEBT] :
            ( ( B = aa_VEB5832953974749077948T_VEBT(F,X) )
           => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2)) ) ) ).

% imageE
tff(fact_5818_imageE,axiom,
    ! [B: vEBT_VEBT,F: fun_real_VEBT_VEBT,A2: set_real] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),image_real_VEBT_VEBT(F,A2)))
     => ~ ! [X: real] :
            ( ( B = aa_real_VEBT_VEBT(F,X) )
           => ~ pp(aa_set_real_bool(member_real(X),A2)) ) ) ).

% imageE
tff(fact_5819_imageE,axiom,
    ! [B: vEBT_VEBT,F: fun_int_VEBT_VEBT,A2: set_int] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),image_int_VEBT_VEBT(F,A2)))
     => ~ ! [X: int] :
            ( ( B = aa_int_VEBT_VEBT(F,X) )
           => ~ pp(aa_set_int_bool(member_int(X),A2)) ) ) ).

% imageE
tff(fact_5820_imageE,axiom,
    ! [B: real,F: fun_nat_real,A2: set_nat] :
      ( pp(aa_set_real_bool(member_real(B),aa_set_nat_set_real(image_nat_real(F),A2)))
     => ~ ! [X: nat] :
            ( ( B = aa_nat_real(F,X) )
           => ~ pp(aa_set_nat_bool(member_nat(X),A2)) ) ) ).

% imageE
tff(fact_5821_imageE,axiom,
    ! [B: real,F: fun_VEBT_VEBT_real,A2: set_VEBT_VEBT] :
      ( pp(aa_set_real_bool(member_real(B),image_VEBT_VEBT_real(F,A2)))
     => ~ ! [X: vEBT_VEBT] :
            ( ( B = aa_VEBT_VEBT_real(F,X) )
           => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2)) ) ) ).

% imageE
tff(fact_5822_image__image,axiom,
    ! [F: fun_real_real,G: fun_nat_real,A2: set_nat] : image_real_real(F,aa_set_nat_set_real(image_nat_real(G),A2)) = aa_set_nat_set_real(image_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_th(F),G)),A2) ).

% image_image
tff(fact_5823_image__image,axiom,
    ! [F: fun_real_nat,G: fun_nat_real,A2: set_nat] : image_real_nat(F,aa_set_nat_set_real(image_nat_real(G),A2)) = aa_set_nat_set_nat(image_nat_nat(aa_fun8286852990929645268at_nat(aTP_Lamm_ti(F),G)),A2) ).

% image_image
tff(fact_5824_image__image,axiom,
    ! [F: fun_real_int,G: fun_nat_real,A2: set_nat] : image_real_int(F,aa_set_nat_set_real(image_nat_real(G),A2)) = aa_set_nat_set_int(image_nat_int(aa_fun4109001971420448560at_int(aTP_Lamm_tj(F),G)),A2) ).

% image_image
tff(fact_5825_image__image,axiom,
    ! [F: fun_int_real,G: fun_nat_int,A2: set_nat] : image_int_real(F,aa_set_nat_set_int(image_nat_int(G),A2)) = aa_set_nat_set_real(image_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_tk(F),G)),A2) ).

% image_image
tff(fact_5826_image__image,axiom,
    ! [F: fun_int_nat,G: fun_nat_int,A2: set_nat] : image_int_nat(F,aa_set_nat_set_int(image_nat_int(G),A2)) = aa_set_nat_set_nat(image_nat_nat(aa_fun519522104774525396at_nat(aTP_Lamm_tl(F),G)),A2) ).

% image_image
tff(fact_5827_image__image,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat,A2: set_nat] : aa_set_nat_set_real(image_nat_real(F),aa_set_nat_set_nat(image_nat_nat(G),A2)) = aa_set_nat_set_real(image_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_tm(F),G)),A2) ).

% image_image
tff(fact_5828_image__image,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A2: set_nat] : aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(image_nat_nat(G),A2)) = aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tn(F),G)),A2) ).

% image_image
tff(fact_5829_image__image,axiom,
    ! [F: fun_nat_int,G: fun_int_nat,A2: set_int] : aa_set_nat_set_int(image_nat_int(F),image_int_nat(G,A2)) = aa_set_int_set_int(image_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_to(F),G)),A2) ).

% image_image
tff(fact_5830_image__image,axiom,
    ! [F: fun_nat_int,G: fun_nat_nat,A2: set_nat] : aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(image_nat_nat(G),A2)) = aa_set_nat_set_int(image_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_tp(F),G)),A2) ).

% image_image
tff(fact_5831_image__image,axiom,
    ! [F: fun_int_int,G: fun_nat_int,A2: set_nat] : aa_set_int_set_int(image_int_int(F),aa_set_nat_set_int(image_nat_int(G),A2)) = aa_set_nat_set_int(image_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_tq(F),G)),A2) ).

% image_image
tff(fact_5832_Compr__image__eq,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,A2: set_VEBT_VEBT,P: fun_VEBT_VEBT_bool] : collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aa_set5278205761110950305T_bool(aTP_Lamm_tr(F),A2),P)) = image_3375948659692109573T_VEBT(F,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aa_set5278205761110950305T_bool(aTP_Lamm_ts(F),A2),P))) ).

% Compr_image_eq
tff(fact_5833_Compr__image__eq,axiom,
    ! [F: fun_real_VEBT_VEBT,A2: set_real,P: fun_VEBT_VEBT_bool] : collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aa_set6789171798632725353T_bool(aTP_Lamm_tt(F),A2),P)) = image_real_VEBT_VEBT(F,collect_real(aa_fun2240483520789045728l_bool(aa_set4656197680294135511l_bool(aTP_Lamm_tu(F),A2),P))) ).

% Compr_image_eq
tff(fact_5834_Compr__image__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,A2: set_VEBT_VEBT,P: fun_real_bool] : collect_real(aa_fun6476784834714468668l_bool(aa_set4098556036960817563l_bool(aTP_Lamm_tv(F),A2),P)) = image_VEBT_VEBT_real(F,collect_VEBT_VEBT(aa_fun2927746532475200174T_bool(aa_set655613085781954469T_bool(aTP_Lamm_tw(F),A2),P))) ).

% Compr_image_eq
tff(fact_5835_Compr__image__eq,axiom,
    ! [F: fun_real_real,A2: set_real,P: fun_real_bool] : collect_real(aa_fun6476784834714468668l_bool(aa_set1278274404314589027l_bool(aTP_Lamm_tx(F),A2),P)) = image_real_real(F,collect_real(aa_fun6476784834714468668l_bool(aa_set1278274404314589027l_bool(aTP_Lamm_ty(F),A2),P))) ).

% Compr_image_eq
tff(fact_5836_Compr__image__eq,axiom,
    ! [F: fun_co4340422964966740447T_VEBT,A2: set_complex,P: fun_VEBT_VEBT_bool] : collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aa_set1540900332140587627T_bool(aTP_Lamm_tz(F),A2),P)) = image_932796090930683071T_VEBT(F,collect_complex(aa_fun3721220998789367262x_bool(aa_set4710227632152643155x_bool(aTP_Lamm_ua(F),A2),P))) ).

% Compr_image_eq
tff(fact_5837_Compr__image__eq,axiom,
    ! [F: fun_complex_real,A2: set_complex,P: fun_real_bool] : collect_real(aa_fun6476784834714468668l_bool(aa_set6829441243277608293l_bool(aTP_Lamm_ub(F),A2),P)) = image_complex_real(F,collect_complex(aa_fun3481569711432096570x_bool(aa_set768065858760106851x_bool(aTP_Lamm_uc(F),A2),P))) ).

% Compr_image_eq
tff(fact_5838_Compr__image__eq,axiom,
    ! [F: fun_nat_VEBT_VEBT,A2: set_nat,P: fun_VEBT_VEBT_bool] : collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aa_set5197589798974504589T_bool(aTP_Lamm_ud(F),A2),P)) = image_nat_VEBT_VEBT(F,collect_nat(aa_fun8987042180644983100t_bool(aa_set2270706540308894223t_bool(aTP_Lamm_ue(F),A2),P))) ).

% Compr_image_eq
tff(fact_5839_Compr__image__eq,axiom,
    ! [F: fun_nat_real,A2: set_nat,P: fun_real_bool] : collect_real(aa_fun6476784834714468668l_bool(aa_set512738241545386119l_bool(aTP_Lamm_uf(F),A2),P)) = aa_set_nat_set_real(image_nat_real(F),collect_nat(aa_fun6526404207338086040t_bool(aa_set4447418679231215587t_bool(aTP_Lamm_ug(F),A2),P))) ).

% Compr_image_eq
tff(fact_5840_Compr__image__eq,axiom,
    ! [F: fun_int_VEBT_VEBT,A2: set_int,P: fun_VEBT_VEBT_bool] : collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aa_set6809977906677068521T_bool(aTP_Lamm_uh(F),A2),P)) = image_int_VEBT_VEBT(F,collect_int(aa_fun5862751214176358880t_bool(aa_set4588796106636019031t_bool(aTP_Lamm_ui(F),A2),P))) ).

% Compr_image_eq
tff(fact_5841_Compr__image__eq,axiom,
    ! [F: fun_int_real,A2: set_int,P: fun_real_bool] : collect_real(aa_fun6476784834714468668l_bool(aa_set8393313469388136163l_bool(aTP_Lamm_uj(F),A2),P)) = image_int_real(F,collect_int(aa_fun3402113240869461820t_bool(aa_set559342874640923363t_bool(aTP_Lamm_uk(F),A2),P))) ).

% Compr_image_eq
tff(fact_5842_zip__inj,axiom,
    ! [A: list_real,B: list_real,A5: list_real,B5: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,A) = aa_list_real_nat(size_size_list_real,B) )
     => ( ( aa_list_real_nat(size_size_list_real,A5) = aa_list_real_nat(size_size_list_real,B5) )
       => ( ( zip_real_real(A,B) = zip_real_real(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5843_zip__inj,axiom,
    ! [A: list_real,B: list_bool,A5: list_real,B5: list_bool] :
      ( ( aa_list_real_nat(size_size_list_real,A) = aa_list_bool_nat(size_size_list_bool,B) )
     => ( ( aa_list_real_nat(size_size_list_real,A5) = aa_list_bool_nat(size_size_list_bool,B5) )
       => ( ( zip_real_bool(A,B) = zip_real_bool(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5844_zip__inj,axiom,
    ! [A: list_real,B: list_nat,A5: list_real,B5: list_nat] :
      ( ( aa_list_real_nat(size_size_list_real,A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( aa_list_real_nat(size_size_list_real,A5) = aa_list_nat_nat(size_size_list_nat,B5) )
       => ( ( zip_real_nat(A,B) = zip_real_nat(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5845_zip__inj,axiom,
    ! [A: list_real,B: list_int,A5: list_real,B5: list_int] :
      ( ( aa_list_real_nat(size_size_list_real,A) = aa_list_int_nat(size_size_list_int,B) )
     => ( ( aa_list_real_nat(size_size_list_real,A5) = aa_list_int_nat(size_size_list_int,B5) )
       => ( ( zip_real_int(A,B) = zip_real_int(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5846_zip__inj,axiom,
    ! [A: list_bool,B: list_real,A5: list_bool,B5: list_real] :
      ( ( aa_list_bool_nat(size_size_list_bool,A) = aa_list_real_nat(size_size_list_real,B) )
     => ( ( aa_list_bool_nat(size_size_list_bool,A5) = aa_list_real_nat(size_size_list_real,B5) )
       => ( ( zip_bool_real(A,B) = zip_bool_real(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5847_zip__inj,axiom,
    ! [A: list_bool,B: list_bool,A5: list_bool,B5: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,A) = aa_list_bool_nat(size_size_list_bool,B) )
     => ( ( aa_list_bool_nat(size_size_list_bool,A5) = aa_list_bool_nat(size_size_list_bool,B5) )
       => ( ( zip_bool_bool(A,B) = zip_bool_bool(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5848_zip__inj,axiom,
    ! [A: list_bool,B: list_nat,A5: list_bool,B5: list_nat] :
      ( ( aa_list_bool_nat(size_size_list_bool,A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( aa_list_bool_nat(size_size_list_bool,A5) = aa_list_nat_nat(size_size_list_nat,B5) )
       => ( ( zip_bool_nat(A,B) = zip_bool_nat(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5849_zip__inj,axiom,
    ! [A: list_bool,B: list_int,A5: list_bool,B5: list_int] :
      ( ( aa_list_bool_nat(size_size_list_bool,A) = aa_list_int_nat(size_size_list_int,B) )
     => ( ( aa_list_bool_nat(size_size_list_bool,A5) = aa_list_int_nat(size_size_list_int,B5) )
       => ( ( zip_bool_int(A,B) = zip_bool_int(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5850_zip__inj,axiom,
    ! [A: list_nat,B: list_real,A5: list_nat,B5: list_real] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = aa_list_real_nat(size_size_list_real,B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A5) = aa_list_real_nat(size_size_list_real,B5) )
       => ( ( zip_nat_real(A,B) = zip_nat_real(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5851_zip__inj,axiom,
    ! [A: list_nat,B: list_bool,A5: list_nat,B5: list_bool] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = aa_list_bool_nat(size_size_list_bool,B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A5) = aa_list_bool_nat(size_size_list_bool,B5) )
       => ( ( zip_nat_bool(A,B) = zip_nat_bool(A5,B5) )
         => ( ( A = A5 )
            & ( B = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5852_all__subset__image,axiom,
    ! [F: fun_nat_real,A2: set_nat,P: fun_set_real_bool] :
      ( ! [B7: set_real] :
          ( pp(aa_set_real_bool(ord_less_eq_set_real(B7),aa_set_nat_set_real(image_nat_real(F),A2)))
         => pp(aa_set_real_bool(P,B7)) )
    <=> ! [B7: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2))
         => pp(aa_set_real_bool(P,aa_set_nat_set_real(image_nat_real(F),B7))) ) ) ).

% all_subset_image
tff(fact_5853_all__subset__image,axiom,
    ! [F: fun_nat_set_nat,A2: set_nat,P: fun_set_set_nat_bool] :
      ( ! [B7: set_set_nat] :
          ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(B7),aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)))
         => pp(aa_set_set_nat_bool(P,B7)) )
    <=> ! [B7: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2))
         => pp(aa_set_set_nat_bool(P,aa_set7246812951530217472et_nat(image_nat_set_nat(F),B7))) ) ) ).

% all_subset_image
tff(fact_5854_all__subset__image,axiom,
    ! [F: fun_nat_nat,A2: set_nat,P: fun_set_nat_bool] :
      ( ! [B7: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),aa_set_nat_set_nat(image_nat_nat(F),A2)))
         => pp(aa_set_nat_bool(P,B7)) )
    <=> ! [B7: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2))
         => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(image_nat_nat(F),B7))) ) ) ).

% all_subset_image
tff(fact_5855_all__subset__image,axiom,
    ! [F: fun_nat_int,A2: set_nat,P: fun_set_int_bool] :
      ( ! [B7: set_int] :
          ( pp(aa_set_int_bool(ord_less_eq_set_int(B7),aa_set_nat_set_int(image_nat_int(F),A2)))
         => pp(aa_set_int_bool(P,B7)) )
    <=> ! [B7: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2))
         => pp(aa_set_int_bool(P,aa_set_nat_set_int(image_nat_int(F),B7))) ) ) ).

% all_subset_image
tff(fact_5856_all__subset__image,axiom,
    ! [F: fun_int_int,A2: set_int,P: fun_set_int_bool] :
      ( ! [B7: set_int] :
          ( pp(aa_set_int_bool(ord_less_eq_set_int(B7),aa_set_int_set_int(image_int_int(F),A2)))
         => pp(aa_set_int_bool(P,B7)) )
    <=> ! [B7: set_int] :
          ( pp(aa_set_int_bool(ord_less_eq_set_int(B7),A2))
         => pp(aa_set_int_bool(P,aa_set_int_set_int(image_int_int(F),B7))) ) ) ).

% all_subset_image
tff(fact_5857_subset__image__iff,axiom,
    ! [B3: set_real,F: fun_nat_real,A2: set_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(B3),aa_set_nat_set_real(image_nat_real(F),A2)))
    <=> ? [AA: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,AA),A2))
          & ( B3 = aa_set_nat_set_real(image_nat_real(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_5858_subset__image__iff,axiom,
    ! [B3: set_set_nat,F: fun_nat_set_nat,A2: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(B3),aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)))
    <=> ? [AA: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,AA),A2))
          & ( B3 = aa_set7246812951530217472et_nat(image_nat_set_nat(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_5859_subset__image__iff,axiom,
    ! [B3: set_nat,F: fun_nat_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),aa_set_nat_set_nat(image_nat_nat(F),A2)))
    <=> ? [AA: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,AA),A2))
          & ( B3 = aa_set_nat_set_nat(image_nat_nat(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_5860_subset__image__iff,axiom,
    ! [B3: set_int,F: fun_nat_int,A2: set_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),aa_set_nat_set_int(image_nat_int(F),A2)))
    <=> ? [AA: set_nat] :
          ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,AA),A2))
          & ( B3 = aa_set_nat_set_int(image_nat_int(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_5861_subset__image__iff,axiom,
    ! [B3: set_int,F: fun_int_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),aa_set_int_set_int(image_int_int(F),A2)))
    <=> ? [AA: set_int] :
          ( pp(aa_set_int_bool(ord_less_eq_set_int(AA),A2))
          & ( B3 = aa_set_int_set_int(image_int_int(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_5862_image__subset__iff,axiom,
    ! [F: fun_nat_nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),A2)),B3))
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_nat_bool(member_nat(aa_nat_nat(F,X4)),B3)) ) ) ).

% image_subset_iff
tff(fact_5863_image__subset__iff,axiom,
    ! [F: fun_nat_real,A2: set_nat,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_set_nat_set_real(image_nat_real(F),A2)),B3))
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_real_bool(member_real(aa_nat_real(F,X4)),B3)) ) ) ).

% image_subset_iff
tff(fact_5864_image__subset__iff,axiom,
    ! [F: fun_nat_set_nat,A2: set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)),B3))
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_set_nat_bool(member_set_nat(aa_nat_set_nat(F,X4)),B3)) ) ) ).

% image_subset_iff
tff(fact_5865_image__subset__iff,axiom,
    ! [F: fun_nat_int,A2: set_nat,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A2)),B3))
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_int_bool(member_int(aa_nat_int(F,X4)),B3)) ) ) ).

% image_subset_iff
tff(fact_5866_image__subset__iff,axiom,
    ! [F: fun_int_int,A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A2)),B3))
    <=> ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_set_int_bool(member_int(aa_int_int(F,X4)),B3)) ) ) ).

% image_subset_iff
tff(fact_5867_subset__imageE,axiom,
    ! [B3: set_real,F: fun_nat_real,A2: set_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(B3),aa_set_nat_set_real(image_nat_real(F),A2)))
     => ~ ! [C6: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C6),A2))
           => ( B3 != aa_set_nat_set_real(image_nat_real(F),C6) ) ) ) ).

% subset_imageE
tff(fact_5868_subset__imageE,axiom,
    ! [B3: set_set_nat,F: fun_nat_set_nat,A2: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(B3),aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)))
     => ~ ! [C6: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C6),A2))
           => ( B3 != aa_set7246812951530217472et_nat(image_nat_set_nat(F),C6) ) ) ) ).

% subset_imageE
tff(fact_5869_subset__imageE,axiom,
    ! [B3: set_nat,F: fun_nat_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),aa_set_nat_set_nat(image_nat_nat(F),A2)))
     => ~ ! [C6: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C6),A2))
           => ( B3 != aa_set_nat_set_nat(image_nat_nat(F),C6) ) ) ) ).

% subset_imageE
tff(fact_5870_subset__imageE,axiom,
    ! [B3: set_int,F: fun_nat_int,A2: set_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),aa_set_nat_set_int(image_nat_int(F),A2)))
     => ~ ! [C6: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C6),A2))
           => ( B3 != aa_set_nat_set_int(image_nat_int(F),C6) ) ) ) ).

% subset_imageE
tff(fact_5871_subset__imageE,axiom,
    ! [B3: set_int,F: fun_int_int,A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),aa_set_int_set_int(image_int_int(F),A2)))
     => ~ ! [C6: set_int] :
            ( pp(aa_set_int_bool(ord_less_eq_set_int(C6),A2))
           => ( B3 != aa_set_int_set_int(image_int_int(F),C6) ) ) ) ).

% subset_imageE
tff(fact_5872_image__subsetI,axiom,
    ! [A2: set_nat,F: fun_nat_nat,B3: set_nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set_nat_bool(member_nat(aa_nat_nat(F,X)),B3)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),A2)),B3)) ) ).

% image_subsetI
tff(fact_5873_image__subsetI,axiom,
    ! [A2: set_nat,F: fun_nat_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_nat_VEBT_VEBT(F,X)),B3)) )
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(image_nat_VEBT_VEBT(F,A2)),B3)) ) ).

% image_subsetI
tff(fact_5874_image__subsetI,axiom,
    ! [A2: set_nat,F: fun_nat_real,B3: set_real] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set_real_bool(member_real(aa_nat_real(F,X)),B3)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(aa_set_nat_set_real(image_nat_real(F),A2)),B3)) ) ).

% image_subsetI
tff(fact_5875_image__subsetI,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,B3: set_nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_set_nat_bool(member_nat(aa_VEBT_VEBT_nat(F,X)),B3)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,image_VEBT_VEBT_nat(F,A2)),B3)) ) ).

% image_subsetI
tff(fact_5876_image__subsetI,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,B3: set_VEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(F,X)),B3)) )
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(image_3375948659692109573T_VEBT(F,A2)),B3)) ) ).

% image_subsetI
tff(fact_5877_image__subsetI,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,B3: set_real] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(F,X)),B3)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(image_VEBT_VEBT_real(F,A2)),B3)) ) ).

% image_subsetI
tff(fact_5878_image__subsetI,axiom,
    ! [A2: set_real,F: fun_real_nat,B3: set_nat] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set_nat_bool(member_nat(aa_real_nat(F,X)),B3)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,image_real_nat(F,A2)),B3)) ) ).

% image_subsetI
tff(fact_5879_image__subsetI,axiom,
    ! [A2: set_real,F: fun_real_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_real_VEBT_VEBT(F,X)),B3)) )
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(image_real_VEBT_VEBT(F,A2)),B3)) ) ).

% image_subsetI
tff(fact_5880_image__subsetI,axiom,
    ! [A2: set_real,F: fun_real_real,B3: set_real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set_real_bool(member_real(aa_real_real(F,X)),B3)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(image_real_real(F,A2)),B3)) ) ).

% image_subsetI
tff(fact_5881_image__subsetI,axiom,
    ! [A2: set_int,F: fun_int_nat,B3: set_nat] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_set_nat_bool(member_nat(aa_int_nat(F,X)),B3)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,image_int_nat(F,A2)),B3)) ) ).

% image_subsetI
tff(fact_5882_image__mono,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => pp(aa_set_real_bool(ord_less_eq_set_real(aa_set_nat_set_real(image_nat_real(F),A2)),aa_set_nat_set_real(image_nat_real(F),B3))) ) ).

% image_mono
tff(fact_5883_image__mono,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)),aa_set7246812951530217472et_nat(image_nat_set_nat(F),B3))) ) ).

% image_mono
tff(fact_5884_image__mono,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),A2)),aa_set_nat_set_nat(image_nat_nat(F),B3))) ) ).

% image_mono
tff(fact_5885_image__mono,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A2)),aa_set_nat_set_int(image_nat_int(F),B3))) ) ).

% image_mono
tff(fact_5886_image__mono,axiom,
    ! [A2: set_int,B3: set_int,F: fun_int_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A2)),aa_set_int_set_int(image_int_int(F),B3))) ) ).

% image_mono
tff(fact_5887_restrict__map__eq_I2_J,axiom,
    ! [M: fun_nat_option_nat,A2: set_nat,K: nat,V: nat] :
      ( ( aa_nat_option_nat(restrict_map_nat_nat(M,A2),K) = aa_nat_option_nat(some_nat,V) )
    <=> ( ( aa_nat_option_nat(M,K) = aa_nat_option_nat(some_nat,V) )
        & pp(aa_set_nat_bool(member_nat(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5888_restrict__map__eq_I2_J,axiom,
    ! [M: fun_VE8174548813682830439on_nat,A2: set_VEBT_VEBT,K: vEBT_VEBT,V: nat] :
      ( ( aa_VEB7679222860772546456on_nat(restri774867724463461460BT_nat(M,A2),K) = aa_nat_option_nat(some_nat,V) )
    <=> ( ( aa_VEB7679222860772546456on_nat(M,K) = aa_nat_option_nat(some_nat,V) )
        & pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5889_restrict__map__eq_I2_J,axiom,
    ! [M: fun_real_option_nat,A2: set_real,K: real,V: nat] :
      ( ( aa_real_option_nat(restri6827137924477938990al_nat(M,A2),K) = aa_nat_option_nat(some_nat,V) )
    <=> ( ( aa_real_option_nat(M,K) = aa_nat_option_nat(some_nat,V) )
        & pp(aa_set_real_bool(member_real(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5890_restrict__map__eq_I2_J,axiom,
    ! [M: fun_int_option_nat,A2: set_int,K: int,V: nat] :
      ( ( aa_int_option_nat(restrict_map_int_nat(M,A2),K) = aa_nat_option_nat(some_nat,V) )
    <=> ( ( aa_int_option_nat(M,K) = aa_nat_option_nat(some_nat,V) )
        & pp(aa_set_int_bool(member_int(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5891_restrict__map__eq_I2_J,axiom,
    ! [M: fun_nat_option_num,A2: set_nat,K: nat,V: num] :
      ( ( aa_nat_option_num(restrict_map_nat_num(M,A2),K) = aa_num_option_num(some_num,V) )
    <=> ( ( aa_nat_option_num(M,K) = aa_num_option_num(some_num,V) )
        & pp(aa_set_nat_bool(member_nat(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5892_restrict__map__eq_I2_J,axiom,
    ! [M: fun_VE3889768403122177201on_num,A2: set_VEBT_VEBT,K: vEBT_VEBT,V: num] :
      ( ( aa_VEB8387466797721384162on_num(restri6555571547474015902BT_num(M,A2),K) = aa_num_option_num(some_num,V) )
    <=> ( ( aa_VEB8387466797721384162on_num(M,K) = aa_num_option_num(some_num,V) )
        & pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5893_restrict__map__eq_I2_J,axiom,
    ! [M: fun_real_option_num,A2: set_real,K: real,V: num] :
      ( ( aa_real_option_num(restri3384469710633717624al_num(M,A2),K) = aa_num_option_num(some_num,V) )
    <=> ( ( aa_real_option_num(M,K) = aa_num_option_num(some_num,V) )
        & pp(aa_set_real_bool(member_real(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5894_restrict__map__eq_I2_J,axiom,
    ! [M: fun_int_option_num,A2: set_int,K: int,V: num] :
      ( ( aa_int_option_num(restrict_map_int_num(M,A2),K) = aa_num_option_num(some_num,V) )
    <=> ( ( aa_int_option_num(M,K) = aa_num_option_num(some_num,V) )
        & pp(aa_set_int_bool(member_int(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5895_restrict__map__eq_I2_J,axiom,
    ! [M: fun_se8577451574693647267on_nat,A2: set_set_nat,K: set_nat,V: nat] :
      ( ( aa_set283173322331585252on_nat(restri1785250428630319752at_nat(M,A2),K) = aa_nat_option_nat(some_nat,V) )
    <=> ( ( aa_set283173322331585252on_nat(M,K) = aa_nat_option_nat(some_nat,V) )
        & pp(aa_set_set_nat_bool(member_set_nat(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5896_restrict__map__eq_I2_J,axiom,
    ! [M: fun_se4292671164132994029on_num,A2: set_set_nat,K: set_nat,V: num] :
      ( ( aa_set991417259280422958on_num(restri7565954251640874194at_num(M,A2),K) = aa_num_option_num(some_num,V) )
    <=> ( ( aa_set991417259280422958on_num(M,K) = aa_num_option_num(some_num,V) )
        & pp(aa_set_set_nat_bool(member_set_nat(K),A2)) ) ) ).

% restrict_map_eq(2)
tff(fact_5897_total__on__empty,axiom,
    ! [R2: set_Pr6218003697084177305l_real] : total_on_real(bot_bot_set_real,R2) ).

% total_on_empty
tff(fact_5898_total__on__empty,axiom,
    ! [R2: set_Pr8621012493645324487l_bool] : total_on_bool(bot_bot_set_bool,R2) ).

% total_on_empty
tff(fact_5899_total__on__empty,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] : total_on_nat(bot_bot_set_nat,R2) ).

% total_on_empty
tff(fact_5900_total__on__empty,axiom,
    ! [R2: set_Pr958786334691620121nt_int] : total_on_int(bot_bot_set_int,R2) ).

% total_on_empty
tff(fact_5901_pigeonhole__infinite,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,image_VEBT_VEBT_nat(F,A2)))
       => ? [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
            & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aa_fun7053968375109723858T_bool(aTP_Lamm_ul(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5902_pigeonhole__infinite,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,image_real_nat(F,A2)))
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
            & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_rea7809428808904475482l_bool(aa_fun3203339554537555706l_bool(aTP_Lamm_um(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5903_pigeonhole__infinite,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,image_VEBT_VEBT_int(F,A2)))
       => ? [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
            & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aa_fun3060702913319154550T_bool(aTP_Lamm_un(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5904_pigeonhole__infinite,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,image_real_int(F,A2)))
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
            & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_rea7809428808904475482l_bool(aa_fun7220475794303443358l_bool(aTP_Lamm_uo(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5905_pigeonhole__infinite,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,image_3793382806556112285omplex(F,A2)))
       => ? [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
            & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aa_fun5117077068891053428T_bool(aTP_Lamm_up(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5906_pigeonhole__infinite,axiom,
    ! [A2: set_real,F: fun_real_complex] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,image_real_complex(F,A2)))
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
            & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_rea7809428808904475482l_bool(aa_fun3442657443821130908l_bool(aTP_Lamm_uq(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5907_pigeonhole__infinite,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,image_2092689629700589388nteger(F,A2)))
       => ? [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
            & ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_VEB2578974996309632030T_bool(aa_fun1850620383256809093T_bool(aTP_Lamm_ur(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5908_pigeonhole__infinite,axiom,
    ! [A2: set_real,F: fun_re6978948838062957830nteger] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,image_4958697645175560720nteger(F,A2)))
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
            & ~ pp(aa_set_real_bool(finite_finite_real,collect_real(aa_rea7809428808904475482l_bool(aa_fun5856998839660973357l_bool(aTP_Lamm_us(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5909_pigeonhole__infinite,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_real_bool(finite_finite_real,aa_set_nat_set_real(image_nat_real(F),A2)))
       => ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
            & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun2005584043565060658t_bool(aTP_Lamm_ut(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5910_pigeonhole__infinite,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(F),A2)))
       => ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
            & ~ pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aa_fun3777555276967886990t_bool(aTP_Lamm_uu(A2),F),X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5911_image__Collect__subsetI,axiom,
    ! [P: fun_complex_bool,F: fun_complex_nat,B3: set_nat] :
      ( ! [X: complex] :
          ( pp(aa_complex_bool(P,X))
         => pp(aa_set_nat_bool(member_nat(aa_complex_nat(F,X)),B3)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,image_complex_nat(F,collect_complex(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5912_image__Collect__subsetI,axiom,
    ! [P: fun_complex_bool,F: fun_co4340422964966740447T_VEBT,B3: set_VEBT_VEBT] :
      ( ! [X: complex] :
          ( pp(aa_complex_bool(P,X))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_complex_VEBT_VEBT(F,X)),B3)) )
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(image_932796090930683071T_VEBT(F,collect_complex(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5913_image__Collect__subsetI,axiom,
    ! [P: fun_complex_bool,F: fun_complex_real,B3: set_real] :
      ( ! [X: complex] :
          ( pp(aa_complex_bool(P,X))
         => pp(aa_set_real_bool(member_real(aa_complex_real(F,X)),B3)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(image_complex_real(F,collect_complex(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5914_image__Collect__subsetI,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat,B3: set_nat] :
      ( ! [X: nat] :
          ( pp(aa_nat_bool(P,X))
         => pp(aa_set_nat_bool(member_nat(aa_nat_nat(F,X)),B3)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),collect_nat(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5915_image__Collect__subsetI,axiom,
    ! [P: fun_nat_bool,F: fun_nat_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ! [X: nat] :
          ( pp(aa_nat_bool(P,X))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_nat_VEBT_VEBT(F,X)),B3)) )
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(image_nat_VEBT_VEBT(F,collect_nat(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5916_image__Collect__subsetI,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real,B3: set_real] :
      ( ! [X: nat] :
          ( pp(aa_nat_bool(P,X))
         => pp(aa_set_real_bool(member_real(aa_nat_real(F,X)),B3)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(aa_set_nat_set_real(image_nat_real(F),collect_nat(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5917_image__Collect__subsetI,axiom,
    ! [P: fun_int_bool,F: fun_int_nat,B3: set_nat] :
      ( ! [X: int] :
          ( pp(aa_int_bool(P,X))
         => pp(aa_set_nat_bool(member_nat(aa_int_nat(F,X)),B3)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,image_int_nat(F,collect_int(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5918_image__Collect__subsetI,axiom,
    ! [P: fun_int_bool,F: fun_int_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ! [X: int] :
          ( pp(aa_int_bool(P,X))
         => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_int_VEBT_VEBT(F,X)),B3)) )
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(image_int_VEBT_VEBT(F,collect_int(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5919_image__Collect__subsetI,axiom,
    ! [P: fun_int_bool,F: fun_int_real,B3: set_real] :
      ( ! [X: int] :
          ( pp(aa_int_bool(P,X))
         => pp(aa_set_real_bool(member_real(aa_int_real(F,X)),B3)) )
     => pp(aa_set_real_bool(ord_less_eq_set_real(image_int_real(F,collect_int(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5920_image__Collect__subsetI,axiom,
    ! [P: fun_complex_bool,F: fun_complex_int,B3: set_int] :
      ( ! [X: complex] :
          ( pp(aa_complex_bool(P,X))
         => pp(aa_set_int_bool(member_int(aa_complex_int(F,X)),B3)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(image_complex_int(F,collect_complex(P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5921_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),A2))) ).

% zero_notin_Suc_image
tff(fact_5922_all__finite__subset__image,axiom,
    ! [F: fun_nat_real,A2: set_nat,P: fun_set_real_bool] :
      ( ! [B7: set_real] :
          ( ( pp(aa_set_real_bool(finite_finite_real,B7))
            & pp(aa_set_real_bool(ord_less_eq_set_real(B7),aa_set_nat_set_real(image_nat_real(F),A2))) )
         => pp(aa_set_real_bool(P,B7)) )
    <=> ! [B7: set_nat] :
          ( ( pp(aa_set_nat_bool(finite_finite_nat,B7))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2)) )
         => pp(aa_set_real_bool(P,aa_set_nat_set_real(image_nat_real(F),B7))) ) ) ).

% all_finite_subset_image
tff(fact_5923_all__finite__subset__image,axiom,
    ! [F: fun_nat_nat,A2: set_nat,P: fun_set_nat_bool] :
      ( ! [B7: set_nat] :
          ( ( pp(aa_set_nat_bool(finite_finite_nat,B7))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),aa_set_nat_set_nat(image_nat_nat(F),A2))) )
         => pp(aa_set_nat_bool(P,B7)) )
    <=> ! [B7: set_nat] :
          ( ( pp(aa_set_nat_bool(finite_finite_nat,B7))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2)) )
         => pp(aa_set_nat_bool(P,aa_set_nat_set_nat(image_nat_nat(F),B7))) ) ) ).

% all_finite_subset_image
tff(fact_5924_all__finite__subset__image,axiom,
    ! [F: fun_complex_nat,A2: set_complex,P: fun_set_nat_bool] :
      ( ! [B7: set_nat] :
          ( ( pp(aa_set_nat_bool(finite_finite_nat,B7))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),image_complex_nat(F,A2))) )
         => pp(aa_set_nat_bool(P,B7)) )
    <=> ! [B7: set_complex] :
          ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
            & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),A2)) )
         => pp(aa_set_nat_bool(P,image_complex_nat(F,B7))) ) ) ).

% all_finite_subset_image
tff(fact_5925_all__finite__subset__image,axiom,
    ! [F: fun_Code_integer_nat,A2: set_Code_integer,P: fun_set_nat_bool] :
      ( ! [B7: set_nat] :
          ( ( pp(aa_set_nat_bool(finite_finite_nat,B7))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),image_951025933927791156er_nat(F,A2))) )
         => pp(aa_set_nat_bool(P,B7)) )
    <=> ! [B7: set_Code_integer] :
          ( ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
            & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),A2)) )
         => pp(aa_set_nat_bool(P,image_951025933927791156er_nat(F,B7))) ) ) ).

% all_finite_subset_image
tff(fact_5926_all__finite__subset__image,axiom,
    ! [F: fun_nat_complex,A2: set_nat,P: fun_set_complex_bool] :
      ( ! [B7: set_complex] :
          ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
            & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),image_nat_complex(F,A2))) )
         => pp(aa_set_complex_bool(P,B7)) )
    <=> ! [B7: set_nat] :
          ( ( pp(aa_set_nat_bool(finite_finite_nat,B7))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2)) )
         => pp(aa_set_complex_bool(P,image_nat_complex(F,B7))) ) ) ).

% all_finite_subset_image
tff(fact_5927_all__finite__subset__image,axiom,
    ! [F: fun_complex_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( ! [B7: set_complex] :
          ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
            & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),image_1468599708987790691omplex(F,A2))) )
         => pp(aa_set_complex_bool(P,B7)) )
    <=> ! [B7: set_complex] :
          ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
            & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),A2)) )
         => pp(aa_set_complex_bool(P,image_1468599708987790691omplex(F,B7))) ) ) ).

% all_finite_subset_image
tff(fact_5928_all__finite__subset__image,axiom,
    ! [F: fun_Co1900716531650368264omplex,A2: set_Code_integer,P: fun_set_complex_bool] :
      ( ! [B7: set_complex] :
          ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
            & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),image_3397630267976458002omplex(F,A2))) )
         => pp(aa_set_complex_bool(P,B7)) )
    <=> ! [B7: set_Code_integer] :
          ( ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
            & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),A2)) )
         => pp(aa_set_complex_bool(P,image_3397630267976458002omplex(F,B7))) ) ) ).

% all_finite_subset_image
tff(fact_5929_all__finite__subset__image,axiom,
    ! [F: fun_nat_Code_integer,A2: set_nat,P: fun_se2712834123562403997r_bool] :
      ( ! [B7: set_Code_integer] :
          ( ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
            & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),image_1215581382706833972nteger(F,A2))) )
         => pp(aa_set1374477347934578310r_bool(P,B7)) )
    <=> ! [B7: set_nat] :
          ( ( pp(aa_set_nat_bool(finite_finite_nat,B7))
            & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2)) )
         => pp(aa_set1374477347934578310r_bool(P,image_1215581382706833972nteger(F,B7))) ) ) ).

% all_finite_subset_image
tff(fact_5930_all__finite__subset__image,axiom,
    ! [F: fun_co8153928105989009288nteger,A2: set_complex,P: fun_se2712834123562403997r_bool] :
      ( ! [B7: set_Code_integer] :
          ( ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
            & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),image_1994230757181692690nteger(F,A2))) )
         => pp(aa_set1374477347934578310r_bool(P,B7)) )
    <=> ! [B7: set_complex] :
          ( ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
            & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),A2)) )
         => pp(aa_set1374477347934578310r_bool(P,image_1994230757181692690nteger(F,B7))) ) ) ).

% all_finite_subset_image
tff(fact_5931_all__finite__subset__image,axiom,
    ! [F: fun_Co8499238673658658103nteger,A2: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( ! [B7: set_Code_integer] :
          ( ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
            & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),image_4470545334726330049nteger(F,A2))) )
         => pp(aa_set1374477347934578310r_bool(P,B7)) )
    <=> ! [B7: set_Code_integer] :
          ( ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
            & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),A2)) )
         => pp(aa_set1374477347934578310r_bool(P,image_4470545334726330049nteger(F,B7))) ) ) ).

% all_finite_subset_image
tff(fact_5932_ex__finite__subset__image,axiom,
    ! [F: fun_nat_real,A2: set_nat,P: fun_set_real_bool] :
      ( ? [B7: set_real] :
          ( pp(aa_set_real_bool(finite_finite_real,B7))
          & pp(aa_set_real_bool(ord_less_eq_set_real(B7),aa_set_nat_set_real(image_nat_real(F),A2)))
          & pp(aa_set_real_bool(P,B7)) )
    <=> ? [B7: set_nat] :
          ( pp(aa_set_nat_bool(finite_finite_nat,B7))
          & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2))
          & pp(aa_set_real_bool(P,aa_set_nat_set_real(image_nat_real(F),B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5933_ex__finite__subset__image,axiom,
    ! [F: fun_nat_nat,A2: set_nat,P: fun_set_nat_bool] :
      ( ? [B7: set_nat] :
          ( pp(aa_set_nat_bool(finite_finite_nat,B7))
          & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),aa_set_nat_set_nat(image_nat_nat(F),A2)))
          & pp(aa_set_nat_bool(P,B7)) )
    <=> ? [B7: set_nat] :
          ( pp(aa_set_nat_bool(finite_finite_nat,B7))
          & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2))
          & pp(aa_set_nat_bool(P,aa_set_nat_set_nat(image_nat_nat(F),B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5934_ex__finite__subset__image,axiom,
    ! [F: fun_complex_nat,A2: set_complex,P: fun_set_nat_bool] :
      ( ? [B7: set_nat] :
          ( pp(aa_set_nat_bool(finite_finite_nat,B7))
          & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),image_complex_nat(F,A2)))
          & pp(aa_set_nat_bool(P,B7)) )
    <=> ? [B7: set_complex] :
          ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
          & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),A2))
          & pp(aa_set_nat_bool(P,image_complex_nat(F,B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5935_ex__finite__subset__image,axiom,
    ! [F: fun_Code_integer_nat,A2: set_Code_integer,P: fun_set_nat_bool] :
      ( ? [B7: set_nat] :
          ( pp(aa_set_nat_bool(finite_finite_nat,B7))
          & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),image_951025933927791156er_nat(F,A2)))
          & pp(aa_set_nat_bool(P,B7)) )
    <=> ? [B7: set_Code_integer] :
          ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
          & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),A2))
          & pp(aa_set_nat_bool(P,image_951025933927791156er_nat(F,B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5936_ex__finite__subset__image,axiom,
    ! [F: fun_nat_complex,A2: set_nat,P: fun_set_complex_bool] :
      ( ? [B7: set_complex] :
          ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
          & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),image_nat_complex(F,A2)))
          & pp(aa_set_complex_bool(P,B7)) )
    <=> ? [B7: set_nat] :
          ( pp(aa_set_nat_bool(finite_finite_nat,B7))
          & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2))
          & pp(aa_set_complex_bool(P,image_nat_complex(F,B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5937_ex__finite__subset__image,axiom,
    ! [F: fun_complex_complex,A2: set_complex,P: fun_set_complex_bool] :
      ( ? [B7: set_complex] :
          ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
          & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),image_1468599708987790691omplex(F,A2)))
          & pp(aa_set_complex_bool(P,B7)) )
    <=> ? [B7: set_complex] :
          ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
          & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),A2))
          & pp(aa_set_complex_bool(P,image_1468599708987790691omplex(F,B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5938_ex__finite__subset__image,axiom,
    ! [F: fun_Co1900716531650368264omplex,A2: set_Code_integer,P: fun_set_complex_bool] :
      ( ? [B7: set_complex] :
          ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
          & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),image_3397630267976458002omplex(F,A2)))
          & pp(aa_set_complex_bool(P,B7)) )
    <=> ? [B7: set_Code_integer] :
          ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
          & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),A2))
          & pp(aa_set_complex_bool(P,image_3397630267976458002omplex(F,B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5939_ex__finite__subset__image,axiom,
    ! [F: fun_nat_Code_integer,A2: set_nat,P: fun_se2712834123562403997r_bool] :
      ( ? [B7: set_Code_integer] :
          ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
          & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),image_1215581382706833972nteger(F,A2)))
          & pp(aa_set1374477347934578310r_bool(P,B7)) )
    <=> ? [B7: set_nat] :
          ( pp(aa_set_nat_bool(finite_finite_nat,B7))
          & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B7),A2))
          & pp(aa_set1374477347934578310r_bool(P,image_1215581382706833972nteger(F,B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5940_ex__finite__subset__image,axiom,
    ! [F: fun_co8153928105989009288nteger,A2: set_complex,P: fun_se2712834123562403997r_bool] :
      ( ? [B7: set_Code_integer] :
          ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
          & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),image_1994230757181692690nteger(F,A2)))
          & pp(aa_set1374477347934578310r_bool(P,B7)) )
    <=> ? [B7: set_complex] :
          ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B7))
          & pp(aa_set_complex_bool(ord_le211207098394363844omplex(B7),A2))
          & pp(aa_set1374477347934578310r_bool(P,image_1994230757181692690nteger(F,B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5941_ex__finite__subset__image,axiom,
    ! [F: fun_Co8499238673658658103nteger,A2: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( ? [B7: set_Code_integer] :
          ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
          & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),image_4470545334726330049nteger(F,A2)))
          & pp(aa_set1374477347934578310r_bool(P,B7)) )
    <=> ? [B7: set_Code_integer] :
          ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B7))
          & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B7),A2))
          & pp(aa_set1374477347934578310r_bool(P,image_4470545334726330049nteger(F,B7))) ) ) ).

% ex_finite_subset_image
tff(fact_5942_finite__subset__image,axiom,
    ! [B3: set_real,F: fun_nat_real,A2: set_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,B3))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(B3),aa_set_nat_set_real(image_nat_real(F),A2)))
       => ? [C6: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C6),A2))
            & pp(aa_set_nat_bool(finite_finite_nat,C6))
            & ( B3 = aa_set_nat_set_real(image_nat_real(F),C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5943_finite__subset__image,axiom,
    ! [B3: set_nat,F: fun_nat_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),aa_set_nat_set_nat(image_nat_nat(F),A2)))
       => ? [C6: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C6),A2))
            & pp(aa_set_nat_bool(finite_finite_nat,C6))
            & ( B3 = aa_set_nat_set_nat(image_nat_nat(F),C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5944_finite__subset__image,axiom,
    ! [B3: set_nat,F: fun_complex_nat,A2: set_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),image_complex_nat(F,A2)))
       => ? [C6: set_complex] :
            ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(C6),A2))
            & pp(aa_set_complex_bool(finite3207457112153483333omplex,C6))
            & ( B3 = image_complex_nat(F,C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5945_finite__subset__image,axiom,
    ! [B3: set_nat,F: fun_Code_integer_nat,A2: set_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B3))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),image_951025933927791156er_nat(F,A2)))
       => ? [C6: set_Code_integer] :
            ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(C6),A2))
            & pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C6))
            & ( B3 = image_951025933927791156er_nat(F,C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5946_finite__subset__image,axiom,
    ! [B3: set_complex,F: fun_nat_complex,A2: set_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B3),image_nat_complex(F,A2)))
       => ? [C6: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C6),A2))
            & pp(aa_set_nat_bool(finite_finite_nat,C6))
            & ( B3 = image_nat_complex(F,C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5947_finite__subset__image,axiom,
    ! [B3: set_complex,F: fun_complex_complex,A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B3),image_1468599708987790691omplex(F,A2)))
       => ? [C6: set_complex] :
            ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(C6),A2))
            & pp(aa_set_complex_bool(finite3207457112153483333omplex,C6))
            & ( B3 = image_1468599708987790691omplex(F,C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5948_finite__subset__image,axiom,
    ! [B3: set_complex,F: fun_Co1900716531650368264omplex,A2: set_Code_integer] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B3),image_3397630267976458002omplex(F,A2)))
       => ? [C6: set_Code_integer] :
            ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(C6),A2))
            & pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C6))
            & ( B3 = image_3397630267976458002omplex(F,C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5949_finite__subset__image,axiom,
    ! [B3: set_Code_integer,F: fun_nat_Code_integer,A2: set_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B3),image_1215581382706833972nteger(F,A2)))
       => ? [C6: set_nat] :
            ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C6),A2))
            & pp(aa_set_nat_bool(finite_finite_nat,C6))
            & ( B3 = image_1215581382706833972nteger(F,C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5950_finite__subset__image,axiom,
    ! [B3: set_Code_integer,F: fun_co8153928105989009288nteger,A2: set_complex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B3),image_1994230757181692690nteger(F,A2)))
       => ? [C6: set_complex] :
            ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(C6),A2))
            & pp(aa_set_complex_bool(finite3207457112153483333omplex,C6))
            & ( B3 = image_1994230757181692690nteger(F,C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5951_finite__subset__image,axiom,
    ! [B3: set_Code_integer,F: fun_Co8499238673658658103nteger,A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B3),image_4470545334726330049nteger(F,A2)))
       => ? [C6: set_Code_integer] :
            ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(C6),A2))
            & pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,C6))
            & ( B3 = image_4470545334726330049nteger(F,C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5952_finite__surj,axiom,
    ! [A2: set_nat,B3: set_real,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(B3),aa_set_nat_set_real(image_nat_real(F),A2)))
       => pp(aa_set_real_bool(finite_finite_real,B3)) ) ) ).

% finite_surj
tff(fact_5953_finite__surj,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),aa_set_nat_set_nat(image_nat_nat(F),A2)))
       => pp(aa_set_nat_bool(finite_finite_nat,B3)) ) ) ).

% finite_surj
tff(fact_5954_finite__surj,axiom,
    ! [A2: set_nat,B3: set_complex,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B3),image_nat_complex(F,A2)))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,B3)) ) ) ).

% finite_surj
tff(fact_5955_finite__surj,axiom,
    ! [A2: set_nat,B3: set_Code_integer,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B3),image_1215581382706833972nteger(F,A2)))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3)) ) ) ).

% finite_surj
tff(fact_5956_finite__surj,axiom,
    ! [A2: set_int,B3: set_nat,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),image_int_nat(F,A2)))
       => pp(aa_set_nat_bool(finite_finite_nat,B3)) ) ) ).

% finite_surj
tff(fact_5957_finite__surj,axiom,
    ! [A2: set_int,B3: set_complex,F: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B3),image_int_complex(F,A2)))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,B3)) ) ) ).

% finite_surj
tff(fact_5958_finite__surj,axiom,
    ! [A2: set_int,B3: set_Code_integer,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B3),image_1587234942943678608nteger(F,A2)))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3)) ) ) ).

% finite_surj
tff(fact_5959_finite__surj,axiom,
    ! [A2: set_complex,B3: set_nat,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),image_complex_nat(F,A2)))
       => pp(aa_set_nat_bool(finite_finite_nat,B3)) ) ) ).

% finite_surj
tff(fact_5960_finite__surj,axiom,
    ! [A2: set_complex,B3: set_complex,F: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B3),image_1468599708987790691omplex(F,A2)))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,B3)) ) ) ).

% finite_surj
tff(fact_5961_finite__surj,axiom,
    ! [A2: set_complex,B3: set_Code_integer,F: fun_co8153928105989009288nteger] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B3),image_1994230757181692690nteger(F,A2)))
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3)) ) ) ).

% finite_surj
tff(fact_5962_image__diff__subset,axiom,
    ! [F: fun_nat_real,A2: set_nat,B3: set_nat] : pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(aa_set_nat_set_real(image_nat_real(F),A2),aa_set_nat_set_real(image_nat_real(F),B3))),aa_set_nat_set_real(image_nat_real(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3)))) ).

% image_diff_subset
tff(fact_5963_image__diff__subset,axiom,
    ! [F: fun_nat_set_nat,A2: set_nat,B3: set_nat] : pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2),aa_set7246812951530217472et_nat(image_nat_set_nat(F),B3))),aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3)))) ).

% image_diff_subset
tff(fact_5964_image__diff__subset,axiom,
    ! [F: fun_nat_nat,A2: set_nat,B3: set_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(image_nat_nat(F),A2)),aa_set_nat_set_nat(image_nat_nat(F),B3))),aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3)))) ).

% image_diff_subset
tff(fact_5965_image__diff__subset,axiom,
    ! [F: fun_int_int,A2: set_int,B3: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(image_int_int(F),A2)),aa_set_int_set_int(image_int_int(F),B3))),aa_set_int_set_int(image_int_int(F),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),B3)))) ).

% image_diff_subset
tff(fact_5966_image__diff__subset,axiom,
    ! [F: fun_nat_int,A2: set_nat,B3: set_nat] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_nat_set_int(image_nat_int(F),A2)),aa_set_nat_set_int(image_nat_int(F),B3))),aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B3)))) ).

% image_diff_subset
tff(fact_5967_nat__seg__image__imp__finite,axiom,
    ! [A2: set_real,F: fun_nat_real,N: nat] :
      ( ( A2 = aa_set_nat_set_real(image_nat_real(F),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N))) )
     => pp(aa_set_real_bool(finite_finite_real,A2)) ) ).

% nat_seg_image_imp_finite
tff(fact_5968_nat__seg__image__imp__finite,axiom,
    ! [A2: set_set_nat,F: fun_nat_set_nat,N: nat] :
      ( ( A2 = aa_set7246812951530217472et_nat(image_nat_set_nat(F),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N))) )
     => pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2)) ) ).

% nat_seg_image_imp_finite
tff(fact_5969_nat__seg__image__imp__finite,axiom,
    ! [A2: set_nat,F: fun_nat_nat,N: nat] :
      ( ( A2 = aa_set_nat_set_nat(image_nat_nat(F),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N))) )
     => pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ).

% nat_seg_image_imp_finite
tff(fact_5970_nat__seg__image__imp__finite,axiom,
    ! [A2: set_int,F: fun_nat_int,N: nat] :
      ( ( A2 = aa_set_nat_set_int(image_nat_int(F),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N))) )
     => pp(aa_set_int_bool(finite_finite_int,A2)) ) ).

% nat_seg_image_imp_finite
tff(fact_5971_nat__seg__image__imp__finite,axiom,
    ! [A2: set_complex,F: fun_nat_complex,N: nat] :
      ( ( A2 = image_nat_complex(F,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N))) )
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ).

% nat_seg_image_imp_finite
tff(fact_5972_nat__seg__image__imp__finite,axiom,
    ! [A2: set_Code_integer,F: fun_nat_Code_integer,N: nat] :
      ( ( A2 = image_1215581382706833972nteger(F,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N))) )
     => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2)) ) ).

% nat_seg_image_imp_finite
tff(fact_5973_finite__conv__nat__seg__image,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
    <=> ? [N6: nat,F7: fun_nat_real] : A2 = aa_set_nat_set_real(image_nat_real(F7),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_5974_finite__conv__nat__seg__image,axiom,
    ! [A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
    <=> ? [N6: nat,F7: fun_nat_set_nat] : A2 = aa_set7246812951530217472et_nat(image_nat_set_nat(F7),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_5975_finite__conv__nat__seg__image,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
    <=> ? [N6: nat,F7: fun_nat_nat] : A2 = aa_set_nat_set_nat(image_nat_nat(F7),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_5976_finite__conv__nat__seg__image,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
    <=> ? [N6: nat,F7: fun_nat_int] : A2 = aa_set_nat_set_int(image_nat_int(F7),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_5977_finite__conv__nat__seg__image,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
    <=> ? [N6: nat,F7: fun_nat_complex] : A2 = image_nat_complex(F7,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_5978_finite__conv__nat__seg__image,axiom,
    ! [A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
    <=> ? [N6: nat,F7: fun_nat_Code_integer] : A2 = image_1215581382706833972nteger(F7,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_5979_image__constant,axiom,
    ! [X3: nat,A2: set_nat,C: vEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(X3),A2))
     => ( image_nat_VEBT_VEBT(aTP_Lamm_in(C),A2) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C),bot_bo8194388402131092736T_VEBT) ) ) ).

% image_constant
tff(fact_5980_image__constant,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
     => ( image_3375948659692109573T_VEBT(aTP_Lamm_uv(C),A2) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C),bot_bo8194388402131092736T_VEBT) ) ) ).

% image_constant
tff(fact_5981_image__constant,axiom,
    ! [X3: real,A2: set_real,C: vEBT_VEBT] :
      ( pp(aa_set_real_bool(member_real(X3),A2))
     => ( image_real_VEBT_VEBT(aTP_Lamm_ii(C),A2) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C),bot_bo8194388402131092736T_VEBT) ) ) ).

% image_constant
tff(fact_5982_image__constant,axiom,
    ! [X3: int,A2: set_int,C: vEBT_VEBT] :
      ( pp(aa_set_int_bool(member_int(X3),A2))
     => ( image_int_VEBT_VEBT(aTP_Lamm_ip(C),A2) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C),bot_bo8194388402131092736T_VEBT) ) ) ).

% image_constant
tff(fact_5983_image__constant,axiom,
    ! [X3: nat,A2: set_nat,C: real] :
      ( pp(aa_set_nat_bool(member_nat(X3),A2))
     => ( aa_set_nat_set_real(image_nat_real(aTP_Lamm_uw(C)),A2) = aa_set_real_set_real(insert_real(C),bot_bot_set_real) ) ) ).

% image_constant
tff(fact_5984_image__constant,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,C: real] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
     => ( image_VEBT_VEBT_real(aTP_Lamm_ih(C),A2) = aa_set_real_set_real(insert_real(C),bot_bot_set_real) ) ) ).

% image_constant
tff(fact_5985_image__constant,axiom,
    ! [X3: real,A2: set_real,C: real] :
      ( pp(aa_set_real_bool(member_real(X3),A2))
     => ( image_real_real(aTP_Lamm_ux(C),A2) = aa_set_real_set_real(insert_real(C),bot_bot_set_real) ) ) ).

% image_constant
tff(fact_5986_image__constant,axiom,
    ! [X3: int,A2: set_int,C: real] :
      ( pp(aa_set_int_bool(member_int(X3),A2))
     => ( image_int_real(aTP_Lamm_uy(C),A2) = aa_set_real_set_real(insert_real(C),bot_bot_set_real) ) ) ).

% image_constant
tff(fact_5987_image__constant,axiom,
    ! [X3: nat,A2: set_nat,C: bool] :
      ( pp(aa_set_nat_bool(member_nat(X3),A2))
     => ( image_nat_bool(aa_bool_fun_nat_bool(aTP_Lamm_io,C),A2) = aa_set_bool_set_bool(insert_bool(C),bot_bot_set_bool) ) ) ).

% image_constant
tff(fact_5988_image__constant,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,C: bool] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
     => ( image_VEBT_VEBT_bool(aTP_Lamm_uz(C),A2) = aa_set_bool_set_bool(insert_bool(C),bot_bot_set_bool) ) ) ).

% image_constant
tff(fact_5989_image__constant__conv,axiom,
    ! [A2: set_real,C: vEBT_VEBT] :
      ( ( ( A2 = bot_bot_set_real )
       => ( image_real_VEBT_VEBT(aTP_Lamm_ii(C),A2) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( A2 != bot_bot_set_real )
       => ( image_real_VEBT_VEBT(aTP_Lamm_ii(C),A2) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C),bot_bo8194388402131092736T_VEBT) ) ) ) ).

% image_constant_conv
tff(fact_5990_image__constant__conv,axiom,
    ! [A2: set_real,C: real] :
      ( ( ( A2 = bot_bot_set_real )
       => ( image_real_real(aTP_Lamm_ux(C),A2) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_real )
       => ( image_real_real(aTP_Lamm_ux(C),A2) = aa_set_real_set_real(insert_real(C),bot_bot_set_real) ) ) ) ).

% image_constant_conv
tff(fact_5991_image__constant__conv,axiom,
    ! [A2: set_real,C: bool] :
      ( ( ( A2 = bot_bot_set_real )
       => ( image_real_bool(aTP_Lamm_ij(C),A2) = bot_bot_set_bool ) )
      & ( ( A2 != bot_bot_set_real )
       => ( image_real_bool(aTP_Lamm_ij(C),A2) = aa_set_bool_set_bool(insert_bool(C),bot_bot_set_bool) ) ) ) ).

% image_constant_conv
tff(fact_5992_image__constant__conv,axiom,
    ! [A2: set_real,C: nat] :
      ( ( ( A2 = bot_bot_set_real )
       => ( image_real_nat(aTP_Lamm_va(C),A2) = bot_bot_set_nat ) )
      & ( ( A2 != bot_bot_set_real )
       => ( image_real_nat(aTP_Lamm_va(C),A2) = aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat) ) ) ) ).

% image_constant_conv
tff(fact_5993_image__constant__conv,axiom,
    ! [A2: set_real,C: int] :
      ( ( ( A2 = bot_bot_set_real )
       => ( image_real_int(aTP_Lamm_vb(C),A2) = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_real )
       => ( image_real_int(aTP_Lamm_vb(C),A2) = aa_set_int_set_int(insert_int(C),bot_bot_set_int) ) ) ) ).

% image_constant_conv
tff(fact_5994_image__constant__conv,axiom,
    ! [A2: set_bool,C: vEBT_VEBT] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( image_bool_VEBT_VEBT(aTP_Lamm_ik(C),A2) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( image_bool_VEBT_VEBT(aTP_Lamm_ik(C),A2) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C),bot_bo8194388402131092736T_VEBT) ) ) ) ).

% image_constant_conv
tff(fact_5995_image__constant__conv,axiom,
    ! [A2: set_bool,C: real] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( image_bool_real(aTP_Lamm_vc(C),A2) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( image_bool_real(aTP_Lamm_vc(C),A2) = aa_set_real_set_real(insert_real(C),bot_bot_set_real) ) ) ) ).

% image_constant_conv
tff(fact_5996_image__constant__conv,axiom,
    ! [A2: set_bool,C: bool] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( image_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_il,C),A2) = bot_bot_set_bool ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( image_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_il,C),A2) = aa_set_bool_set_bool(insert_bool(C),bot_bot_set_bool) ) ) ) ).

% image_constant_conv
tff(fact_5997_image__constant__conv,axiom,
    ! [A2: set_bool,C: nat] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( image_bool_nat(aTP_Lamm_vd(C),A2) = bot_bot_set_nat ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( image_bool_nat(aTP_Lamm_vd(C),A2) = aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat) ) ) ) ).

% image_constant_conv
tff(fact_5998_image__constant__conv,axiom,
    ! [A2: set_bool,C: int] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( image_bool_int(aTP_Lamm_ve(C),A2) = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( image_bool_int(aTP_Lamm_ve(C),A2) = aa_set_int_set_int(insert_int(C),bot_bot_set_int) ) ) ) ).

% image_constant_conv
tff(fact_5999_zip__same__conv__map,axiom,
    ! [Xs: list_bool] : zip_bool_bool(Xs,Xs) = map_bo2576003307432440276l_bool(aTP_Lamm_nm,Xs) ).

% zip_same_conv_map
tff(fact_6000_zip__same__conv__map,axiom,
    ! [Xs: list_nat] : zip_nat_nat(Xs,Xs) = map_na7298421622053143531at_nat(aTP_Lamm_nn,Xs) ).

% zip_same_conv_map
tff(fact_6001_zip__same__conv__map,axiom,
    ! [Xs: list_int] : zip_int_int(Xs,Xs) = map_in7157766398909135175nt_int(aTP_Lamm_no,Xs) ).

% zip_same_conv_map
tff(fact_6002_zip__same__conv__map,axiom,
    ! [Xs: list_Code_integer] : zip_Co3543743374963494515nteger(Xs,Xs) = map_Co3589949550033412536nteger(aTP_Lamm_np,Xs) ).

% zip_same_conv_map
tff(fact_6003_pair__list__split,axiom,
    ! [L: list_P8689742595348180415l_real] :
      ~ ! [L12: list_real,L23: list_real] :
          ( ( L = zip_real_real(L12,L23) )
         => ( ( aa_list_real_nat(size_size_list_real,L12) = aa_list_real_nat(size_size_list_real,L23) )
           => ( size_s3932428310213730859l_real(L) != aa_list_real_nat(size_size_list_real,L23) ) ) ) ).

% pair_list_split
tff(fact_6004_pair__list__split,axiom,
    ! [L: list_P3020538015149218416l_bool] :
      ~ ! [L12: list_real,L23: list_bool] :
          ( ( L = zip_real_bool(L12,L23) )
         => ( ( aa_list_real_nat(size_size_list_real,L12) = aa_list_bool_nat(size_size_list_bool,L23) )
           => ( size_s2850477280887376644l_bool(L) != aa_list_bool_nat(size_size_list_bool,L23) ) ) ) ).

% pair_list_split
tff(fact_6005_pair__list__split,axiom,
    ! [L: list_P6834414599653733731al_nat] :
      ~ ! [L12: list_real,L23: list_nat] :
          ( ( L = zip_real_nat(L12,L23) )
         => ( ( aa_list_real_nat(size_size_list_real,L12) = aa_list_nat_nat(size_size_list_nat,L23) )
           => ( size_s1877336372972134351al_nat(L) != aa_list_nat_nat(size_size_list_nat,L23) ) ) ) ).

% pair_list_split
tff(fact_6006_pair__list__split,axiom,
    ! [L: list_P4344331454722006975al_int] :
      ~ ! [L12: list_real,L23: list_int] :
          ( ( L = zip_real_int(L12,L23) )
         => ( ( aa_list_real_nat(size_size_list_real,L12) = aa_list_int_nat(size_size_list_int,L23) )
           => ( size_s8610625264895183403al_int(L) != aa_list_int_nat(size_size_list_int,L23) ) ) ) ).

% pair_list_split
tff(fact_6007_pair__list__split,axiom,
    ! [L: list_P8975979458798387586l_real] :
      ~ ! [L12: list_bool,L23: list_real] :
          ( ( L = zip_bool_real(L12,L23) )
         => ( ( aa_list_bool_nat(size_size_list_bool,L12) = aa_list_real_nat(size_size_list_real,L23) )
           => ( size_s8805918724536545814l_real(L) != aa_list_real_nat(size_size_list_real,L23) ) ) ) ).

% pair_list_split
tff(fact_6008_pair__list__split,axiom,
    ! [L: list_P2722043638750504429l_bool] :
      ~ ! [L12: list_bool,L23: list_bool] :
          ( ( L = zip_bool_bool(L12,L23) )
         => ( ( aa_list_bool_nat(size_size_list_bool,L12) = aa_list_bool_nat(size_size_list_bool,L23) )
           => ( size_s6988337448923680857l_bool(L) != aa_list_bool_nat(size_size_list_bool,L23) ) ) ) ).

% pair_list_split
tff(fact_6009_pair__list__split,axiom,
    ! [L: list_P18149847613930150ol_nat] :
      ~ ! [L12: list_bool,L23: list_nat] :
          ( ( L = zip_bool_nat(L12,L23) )
         => ( ( aa_list_bool_nat(size_size_list_bool,L12) = aa_list_nat_nat(size_size_list_nat,L23) )
           => ( size_s3243772863075988794ol_nat(L) != aa_list_nat_nat(size_size_list_nat,L23) ) ) ) ).

% pair_list_split
tff(fact_6010_pair__list__split,axiom,
    ! [L: list_P6751438739536979202ol_int] :
      ~ ! [L12: list_bool,L23: list_int] :
          ( ( L = zip_bool_int(L12,L23) )
         => ( ( aa_list_bool_nat(size_size_list_bool,L12) = aa_list_int_nat(size_size_list_int,L23) )
           => ( size_s753689718144262038ol_int(L) != aa_list_int_nat(size_size_list_int,L23) ) ) ) ).

% pair_list_split
tff(fact_6011_pair__list__split,axiom,
    ! [L: list_P3644420460460130531t_real] :
      ~ ! [L12: list_nat,L23: list_real] :
          ( ( L = zip_nat_real(L12,L23) )
         => ( ( aa_list_nat_nat(size_size_list_nat,L12) = aa_list_real_nat(size_size_list_real,L23) )
           => ( size_s7910714270633306959t_real(L) != aa_list_real_nat(size_size_list_real,L23) ) ) ) ).

% pair_list_split
tff(fact_6012_pair__list__split,axiom,
    ! [L: list_P6781292141273955916t_bool] :
      ~ ! [L12: list_nat,L23: list_bool] :
          ( ( L = zip_nat_bool(L12,L23) )
         => ( ( aa_list_nat_nat(size_size_list_nat,L12) = aa_list_bool_nat(size_size_list_bool,L23) )
           => ( size_s783543119881238752t_bool(L) != aa_list_bool_nat(size_size_list_bool,L23) ) ) ) ).

% pair_list_split
tff(fact_6013_the__elem__image__unique,axiom,
    ! [A2: set_nat,F: fun_nat_real,X3: nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),A2))
           => ( aa_nat_real(F,Y3) = aa_nat_real(F,X3) ) )
       => ( the_elem_real(aa_set_nat_set_real(image_nat_real(F),A2)) = aa_nat_real(F,X3) ) ) ) ).

% the_elem_image_unique
tff(fact_6014_the__elem__image__unique,axiom,
    ! [A2: set_nat,F: fun_nat_set_nat,X3: nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),A2))
           => ( aa_nat_set_nat(F,Y3) = aa_nat_set_nat(F,X3) ) )
       => ( the_elem_set_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)) = aa_nat_set_nat(F,X3) ) ) ) ).

% the_elem_image_unique
tff(fact_6015_the__elem__image__unique,axiom,
    ! [A2: set_nat,F: fun_nat_nat,X3: nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),A2))
           => ( aa_nat_nat(F,Y3) = aa_nat_nat(F,X3) ) )
       => ( aa_set_nat_nat(the_elem_nat,aa_set_nat_set_nat(image_nat_nat(F),A2)) = aa_nat_nat(F,X3) ) ) ) ).

% the_elem_image_unique
tff(fact_6016_the__elem__image__unique,axiom,
    ! [A2: set_nat,F: fun_nat_int,X3: nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),A2))
           => ( aa_nat_int(F,Y3) = aa_nat_int(F,X3) ) )
       => ( aa_set_int_int(the_elem_int,aa_set_nat_set_int(image_nat_int(F),A2)) = aa_nat_int(F,X3) ) ) ) ).

% the_elem_image_unique
tff(fact_6017_the__elem__image__unique,axiom,
    ! [A2: set_int,F: fun_int_int,X3: int] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [Y3: int] :
            ( pp(aa_set_int_bool(member_int(Y3),A2))
           => ( aa_int_int(F,Y3) = aa_int_int(F,X3) ) )
       => ( aa_set_int_int(the_elem_int,aa_set_int_set_int(image_int_int(F),A2)) = aa_int_int(F,X3) ) ) ) ).

% the_elem_image_unique
tff(fact_6018_restrict__map__upd,axiom,
    ! [F: fun_nat_option_nat,S3: set_nat,K: nat,V: nat] : fun_up1493157387958331631on_nat(restrict_map_nat_nat(F,S3),K,aa_nat_option_nat(some_nat,V)) = restrict_map_nat_nat(fun_up1493157387958331631on_nat(F,K,aa_nat_option_nat(some_nat,V)),aa_set_nat_set_nat(insert_nat(K),S3)) ).

% restrict_map_upd
tff(fact_6019_restrict__map__upd,axiom,
    ! [F: fun_VE8174548813682830439on_nat,S3: set_VEBT_VEBT,K: vEBT_VEBT,V: nat] : fun_up5885881570350532375on_nat(restri774867724463461460BT_nat(F,S3),K,aa_nat_option_nat(some_nat,V)) = restri774867724463461460BT_nat(fun_up5885881570350532375on_nat(F,K,aa_nat_option_nat(some_nat,V)),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(K),S3)) ).

% restrict_map_upd
tff(fact_6020_restrict__map__upd,axiom,
    ! [F: fun_int_option_nat,S3: set_int,K: int,V: nat] : fun_up3620524117960394059on_nat(restrict_map_int_nat(F,S3),K,aa_nat_option_nat(some_nat,V)) = restrict_map_int_nat(fun_up3620524117960394059on_nat(F,K,aa_nat_option_nat(some_nat,V)),aa_set_int_set_int(insert_int(K),S3)) ).

% restrict_map_upd
tff(fact_6021_restrict__map__upd,axiom,
    ! [F: fun_bool_option_nat,S3: set_bool,K: bool,V: nat] : fun_up4506005684929255546on_nat(restri5405821841207159351ol_nat(F,S3),K,aa_nat_option_nat(some_nat,V)) = restri5405821841207159351ol_nat(fun_up4506005684929255546on_nat(F,K,aa_nat_option_nat(some_nat,V)),aa_set_bool_set_bool(insert_bool(K),S3)) ).

% restrict_map_upd
tff(fact_6022_restrict__map__upd,axiom,
    ! [F: fun_real_option_nat,S3: set_real,K: real,V: nat] : fun_up6677080212936659659on_nat(restri6827137924477938990al_nat(F,S3),K,aa_nat_option_nat(some_nat,V)) = restri6827137924477938990al_nat(fun_up6677080212936659659on_nat(F,K,aa_nat_option_nat(some_nat,V)),aa_set_real_set_real(insert_real(K),S3)) ).

% restrict_map_upd
tff(fact_6023_restrict__map__upd,axiom,
    ! [F: fun_nat_option_num,S3: set_nat,K: nat,V: num] : fun_up2201401324907169337on_num(restrict_map_nat_num(F,S3),K,aa_num_option_num(some_num,V)) = restrict_map_nat_num(fun_up2201401324907169337on_num(F,K,aa_num_option_num(some_num,V)),aa_set_nat_set_nat(insert_nat(K),S3)) ).

% restrict_map_upd
tff(fact_6024_restrict__map__upd,axiom,
    ! [F: fun_VE3889768403122177201on_num,S3: set_VEBT_VEBT,K: vEBT_VEBT,V: num] : fun_up6594125507299370081on_num(restri6555571547474015902BT_num(F,S3),K,aa_num_option_num(some_num,V)) = restri6555571547474015902BT_num(fun_up6594125507299370081on_num(F,K,aa_num_option_num(some_num,V)),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(K),S3)) ).

% restrict_map_upd
tff(fact_6025_restrict__map__upd,axiom,
    ! [F: fun_int_option_num,S3: set_int,K: int,V: num] : fun_up4328768054909231765on_num(restrict_map_int_num(F,S3),K,aa_num_option_num(some_num,V)) = restrict_map_int_num(fun_up4328768054909231765on_num(F,K,aa_num_option_num(some_num,V)),aa_set_int_set_int(insert_int(K),S3)) ).

% restrict_map_upd
tff(fact_6026_restrict__map__upd,axiom,
    ! [F: fun_bool_option_num,S3: set_bool,K: bool,V: num] : fun_up5214249621878093252on_num(restri1963153627362937985ol_num(F,S3),K,aa_num_option_num(some_num,V)) = restri1963153627362937985ol_num(fun_up5214249621878093252on_num(F,K,aa_num_option_num(some_num,V)),aa_set_bool_set_bool(insert_bool(K),S3)) ).

% restrict_map_upd
tff(fact_6027_restrict__map__upd,axiom,
    ! [F: fun_real_option_num,S3: set_real,K: real,V: num] : fun_up7385324149885497365on_num(restri3384469710633717624al_num(F,S3),K,aa_num_option_num(some_num,V)) = restri3384469710633717624al_num(fun_up7385324149885497365on_num(F,K,aa_num_option_num(some_num,V)),aa_set_real_set_real(insert_real(K),S3)) ).

% restrict_map_upd
tff(fact_6028_graph__restrictD_I2_J,axiom,
    ! [K: int,V: int,M: fun_int_option_int,A2: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(K),V),graph_int_int(restrict_map_int_int(M,A2)))
     => ( aa_int_option_int(M,K) = aa_int_option_int(some_int,V) ) ) ).

% graph_restrictD(2)
tff(fact_6029_graph__restrictD_I2_J,axiom,
    ! [K: code_integer,V: code_integer,M: fun_Co8029881127924365703nteger,A2: set_Code_integer] :
      ( member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(K),V),graph_5282091004195177018nteger(restri7010877615957751788nteger(M,A2)))
     => ( aa_Cod5017737808611256136nteger(M,K) = aa_Cod5017737808611256136nteger(some_Code_integer,V) ) ) ).

% graph_restrictD(2)
tff(fact_6030_graph__restrictD_I2_J,axiom,
    ! [K: nat,V: nat,M: fun_nat_option_nat,A2: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),V),graph_nat_nat(restrict_map_nat_nat(M,A2)))
     => ( aa_nat_option_nat(M,K) = aa_nat_option_nat(some_nat,V) ) ) ).

% graph_restrictD(2)
tff(fact_6031_graph__restrictD_I2_J,axiom,
    ! [K: vEBT_VEBT,V: nat,M: fun_VE8174548813682830439on_nat,A2: set_VEBT_VEBT] :
      ( member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(K),V),graph_VEBT_VEBT_nat(restri774867724463461460BT_nat(M,A2)))
     => ( aa_VEB7679222860772546456on_nat(M,K) = aa_nat_option_nat(some_nat,V) ) ) ).

% graph_restrictD(2)
tff(fact_6032_graph__restrictD_I2_J,axiom,
    ! [K: nat,V: num,M: fun_nat_option_num,A2: set_nat] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(K),V),graph_nat_num(restrict_map_nat_num(M,A2)))
     => ( aa_nat_option_num(M,K) = aa_num_option_num(some_num,V) ) ) ).

% graph_restrictD(2)
tff(fact_6033_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_1
tff(fact_6034_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_2
tff(fact_6035_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Code_integer,Ys: list_Code_integer,Y: code_integer] :
      ( ( size_s3445333598471063425nteger(Xs) = size_s3445333598471063425nteger(Ys) )
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Y),set_Code_integer2(Ys)))
       => ~ ! [X: code_integer] : ~ member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X),Y),set_Pr920681315882439344nteger(zip_Co3543743374963494515nteger(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6036_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) )
     => ( pp(aa_set_nat_bool(member_nat(Y),aa_list_nat_set_nat(set_nat2,Ys)))
       => ~ ! [X: vEBT_VEBT] : ~ member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X),Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6037_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys)))
       => ~ ! [X: real] : ~ member7262085504369356948T_VEBT(aa_VEB1592731369666541301T_VEBT(produc6931449550656315951T_VEBT(X),Y),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6038_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_real,Y: real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_set_real_bool(member_real(Y),aa_lis3769085826154824284t_real(set_real2,Ys)))
       => ~ ! [X: real] : ~ member7849222048561428706l_real(aa_rea5330357063016736847l_real(produc4511245868158468465l_real(X),Y),set_Pr5999470521830281550l_real(zip_real_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6039_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_bool,Y: bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_set_bool_bool(member_bool(Y),aa_lis3587183946281582166t_bool(set_bool2,Ys)))
       => ~ ! [X: real] : ~ member3295258257541327095l_bool(aa_boo8879657058893838011l_bool(produc3350604258665657362l_bool(X),Y),set_Pr3131006541957657995l_bool(zip_real_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6040_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_nat,Y: nat] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_set_nat_bool(member_nat(Y),aa_list_nat_set_nat(set_nat2,Ys)))
       => ~ ! [X: real] : ~ member5805532792777349510al_nat(aa_nat531201698645514447al_nat(produc3181502643871035669al_nat(X),Y),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6041_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_int,Y: int] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( pp(aa_set_int_bool(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_6042_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Y),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys)))
       => ~ ! [X: bool] : ~ member5465154751058088621T_VEBT(produc9074556814250467700T_VEBT(X,Y),set_Pr5360009138685866265T_VEBT(zip_bool_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6043_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_real,Y: real] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_set_real_bool(member_real(Y),aa_lis3769085826154824284t_real(set_real2,Ys)))
       => ~ ! [X: bool] : ~ member3680586897080482057l_real(aa_rea3799919874630508124l_real(produc7587643690356850540l_real(X),Y),set_Pr3516335181496812957l_real(zip_bool_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6044_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_bool,Ys: list_bool,Y: bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_set_bool_bool(member_bool(Y),aa_lis3587183946281582166t_bool(set_bool2,Ys)))
       => ~ ! [X: bool] : ~ member4869097156692242192l_bool(aa_boo3115168258980966476l_bool(produc8297744024737662807l_bool(X),Y),set_Pr2237862708767489404l_bool(zip_bool_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6045_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Code_integer,Ys: list_Code_integer,X3: code_integer] :
      ( ( size_s3445333598471063425nteger(Xs) = size_s3445333598471063425nteger(Ys) )
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),set_Code_integer2(Xs)))
       => ~ ! [Y3: code_integer] : ~ member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X3),Y3),set_Pr920681315882439344nteger(zip_Co3543743374963494515nteger(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6046_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
       => ~ ! [Y3: real] : ~ member8675245146396747942T_real(aa_rea4189226818901770745T_real(produc8117437818029410057T_real(X3),Y3),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6047_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
       => ~ ! [Y3: bool] : ~ member3504487111852181555T_bool(aa_boo2420671634680568175T_bool(produc3035007008238225530T_bool(X3),Y3),set_Pr3399341499479959199T_bool(zip_VEBT_VEBT_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6048_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
       => ~ ! [Y3: nat] : ~ member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Y3),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6049_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
       => ~ ! [Y3: int] : ~ member5419026705395827622BT_int(aa_int7883028908994967673BT_int(produc736041933913180425BT_int(X3),Y3),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6050_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_real,X3: real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
       => ~ ! [Y3: real] : ~ member7849222048561428706l_real(aa_rea5330357063016736847l_real(produc4511245868158468465l_real(X3),Y3),set_Pr5999470521830281550l_real(zip_real_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6051_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_bool,X3: real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
       => ~ ! [Y3: bool] : ~ member3295258257541327095l_bool(aa_boo8879657058893838011l_bool(produc3350604258665657362l_bool(X3),Y3),set_Pr3131006541957657995l_bool(zip_real_bool(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6052_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_nat,X3: real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
       => ~ ! [Y3: nat] : ~ member5805532792777349510al_nat(aa_nat531201698645514447al_nat(produc3181502643871035669al_nat(X3),Y3),set_Pr3174298344852596722al_nat(zip_real_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6053_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_int,X3: real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,Xs)))
       => ~ ! [Y3: int] : ~ member1627681773268152802al_int(produc3179012173361985393al_int(X3,Y3),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6054_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_bool,Ys: list_real,X3: bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
       => ~ ! [Y3: real] : ~ member3680586897080482057l_real(aa_rea3799919874630508124l_real(produc7587643690356850540l_real(X3),Y3),set_Pr3516335181496812957l_real(zip_bool_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6055_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L4: list_nat,F: fun_nat_real] :
      ( ( aa_list_nat_nat(size_size_list_nat,L) = aa_list_nat_nat(size_size_list_nat,L4) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,L)))
           => ( aa_nat_real(F,aa_nat_nat(nth_nat(L),I2)) = aa_nat_real(F,aa_nat_nat(nth_nat(L4),I2)) ) )
       => ( aa_set_nat_set_real(image_nat_real(F),aa_list_nat_set_nat(set_nat2,L)) = aa_set_nat_set_real(image_nat_real(F),aa_list_nat_set_nat(set_nat2,L4)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_6056_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L4: list_nat,F: fun_nat_set_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,L) = aa_list_nat_nat(size_size_list_nat,L4) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(L4),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,L4)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_6057_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L4: list_nat,F: fun_nat_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,L) = aa_list_nat_nat(size_size_list_nat,L4) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(L4),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,L4)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_6058_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L4: list_nat,F: fun_nat_int] :
      ( ( aa_list_nat_nat(size_size_list_nat,L) = aa_list_nat_nat(size_size_list_nat,L4) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(L4),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,L4)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_6059_set__image__eq__pointwiseI,axiom,
    ! [L: list_int,L4: list_int,F: fun_int_int] :
      ( ( aa_list_int_nat(size_size_list_int,L) = aa_list_int_nat(size_size_list_int,L4) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_list_int_nat(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(L4),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,L4)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_6060_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBTi_nat,X3: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( pp(aa_set_nat_bool(member_nat(aa_VEBT_VEBTi_nat(F,X3)),image_VEBT_VEBTi_nat(F,set_VEBT_VEBTi2(L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6061_in__set__image__conv__nth,axiom,
    ! [F: fun_VE9120655849212850072T_VEBT,X3: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_VEB4941942288398148609T_VEBT(F,X3)),image_7547481670047419768T_VEBT(F,set_VEBT_VEBTi2(L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6062_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBTi_real,X3: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( pp(aa_set_real_bool(member_real(aa_VEBT_VEBTi_real(F,X3)),image_6202559892754154600i_real(F,set_VEBT_VEBTi2(L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6063_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBTi_int,X3: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( pp(aa_set_int_bool(member_int(aa_VEBT_VEBTi_int(F,X3)),image_VEBT_VEBTi_int(F,set_VEBT_VEBTi2(L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6064_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBT_nat,X3: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(aa_VEBT_VEBT_nat(F,X3)),image_VEBT_VEBT_nat(F,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6065_in__set__image__conv__nth,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,X3: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(F,X3)),image_3375948659692109573T_VEBT(F,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6066_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBT_real,X3: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(F,X3)),image_VEBT_VEBT_real(F,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(L)))
          & ( aa_VEBT_VEBT_real(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I3)) = aa_VEBT_VEBT_real(F,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6067_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBT_int,X3: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(F,X3)),image_VEBT_VEBT_int(F,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(L)))
          & ( aa_VEBT_VEBT_int(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I3)) = aa_VEBT_VEBT_int(F,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6068_in__set__image__conv__nth,axiom,
    ! [F: fun_real_nat,X3: real,L: list_real] :
      ( pp(aa_set_nat_bool(member_nat(aa_real_nat(F,X3)),image_real_nat(F,aa_lis3769085826154824284t_real(set_real2,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_real_nat(size_size_list_real,L)))
          & ( aa_real_nat(F,aa_nat_real(nth_real(L),I3)) = aa_real_nat(F,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6069_in__set__image__conv__nth,axiom,
    ! [F: fun_real_VEBT_VEBT,X3: real,L: list_real] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(aa_real_VEBT_VEBT(F,X3)),image_real_VEBT_VEBT(F,aa_lis3769085826154824284t_real(set_real2,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_real_nat(size_size_list_real,L)))
          & ( aa_real_VEBT_VEBT(F,aa_nat_real(nth_real(L),I3)) = aa_real_VEBT_VEBT(F,X3) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6070_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_6071_map__zip1,axiom,
    ! [K: nat,L: list_VEBT_VEBT] : map_VE6666121349297532579BT_nat(aTP_Lamm_vf(K),L) = zip_VEBT_VEBT_nat(L,replicate_nat(size_s6755466524823107622T_VEBT(L),K)) ).

% map_zip1
tff(fact_6072_map__zip1,axiom,
    ! [K: code_integer,L: list_Code_integer] : map_Co3589949550033412536nteger(aTP_Lamm_vg(K),L) = zip_Co3543743374963494515nteger(L,replic7707675349574490269nteger(size_s3445333598471063425nteger(L),K)) ).

% map_zip1
tff(fact_6073_map__zip1,axiom,
    ! [K: vEBT_VEBT,L: list_real] : map_re7205069664741861231T_VEBT(aTP_Lamm_vh(K),L) = zip_real_VEBT_VEBT(L,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_real_nat(size_size_list_real,L)),K)) ).

% map_zip1
tff(fact_6074_map__zip1,axiom,
    ! [K: bool,L: list_real] : map_re2541142195850447826l_bool(aTP_Lamm_vi(K),L) = zip_real_bool(L,replicate_bool(aa_list_real_nat(size_size_list_real,L),K)) ).

% map_zip1
tff(fact_6075_map__zip1,axiom,
    ! [K: vEBT_VEBT,L: list_bool] : map_bo7064569317639751025T_VEBT(aTP_Lamm_vj(K),L) = zip_bool_VEBT_VEBT(L,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_bool_nat(size_size_list_bool,L)),K)) ).

% map_zip1
tff(fact_6076_map__zip1,axiom,
    ! [K: bool,L: list_bool] : map_bo2576003307432440276l_bool(aTP_Lamm_vk(K),L) = zip_bool_bool(L,replicate_bool(aa_list_bool_nat(size_size_list_bool,L),K)) ).

% map_zip1
tff(fact_6077_map__zip1,axiom,
    ! [K: vEBT_VEBT,L: list_nat] : map_na3584885621601055599T_VEBT(aTP_Lamm_vl(K),L) = zip_nat_VEBT_VEBT(L,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_nat_nat(size_size_list_nat,L)),K)) ).

% map_zip1
tff(fact_6078_map__zip1,axiom,
    ! [K: bool,L: list_nat] : map_na1545282596343781330t_bool(aTP_Lamm_vm(K),L) = zip_nat_bool(L,replicate_bool(aa_list_nat_nat(size_size_list_nat,L),K)) ).

% map_zip1
tff(fact_6079_map__zip1,axiom,
    ! [K: nat,L: list_nat] : map_na7298421622053143531at_nat(aTP_Lamm_vn(K),L) = zip_nat_nat(L,replicate_nat(aa_list_nat_nat(size_size_list_nat,L),K)) ).

% map_zip1
tff(fact_6080_map__zip1,axiom,
    ! [K: num,L: list_nat] : map_na8006665559001981237at_num(aTP_Lamm_vo(K),L) = zip_nat_num(L,replicate_num(aa_list_nat_nat(size_size_list_nat,L),K)) ).

% map_zip1
tff(fact_6081_map__zip2,axiom,
    ! [K: code_integer,L: list_Code_integer] : map_Co3589949550033412536nteger(produc1086072967326762835nteger(K),L) = zip_Co3543743374963494515nteger(replic7707675349574490269nteger(size_s3445333598471063425nteger(L),K),L) ).

% map_zip2
tff(fact_6082_map__zip2,axiom,
    ! [K: nat,L: list_num] : map_nu4721551698833171051at_num(product_Pair_nat_num(K),L) = zip_nat_num(replicate_nat(size_size_list_num(L),K),L) ).

% map_zip2
tff(fact_6083_map__zip2,axiom,
    ! [K: vEBT_VEBT,L: list_real] : map_re8618229306769252225T_real(produc8117437818029410057T_real(K),L) = zip_VEBT_VEBT_real(aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_real_nat(size_size_list_real,L)),K),L) ).

% map_zip2
tff(fact_6084_map__zip2,axiom,
    ! [K: bool,L: list_real] : map_re2926470835389602788l_real(produc7587643690356850540l_real(K),L) = zip_bool_real(replicate_bool(aa_list_real_nat(size_size_list_real,L),K),L) ).

% map_zip2
tff(fact_6085_map__zip2,axiom,
    ! [K: vEBT_VEBT,L: list_bool] : map_bo5103901678433843959T_bool(produc3035007008238225530T_bool(K),L) = zip_VEBT_VEBT_bool(aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_bool_nat(size_size_list_bool,L)),K),L) ).

% map_zip2
tff(fact_6086_map__zip2,axiom,
    ! [K: bool,L: list_bool] : map_bo2576003307432440276l_bool(produc8297744024737662807l_bool(K),L) = zip_bool_bool(replicate_bool(aa_list_bool_nat(size_size_list_bool,L),K),L) ).

% map_zip2
tff(fact_6087_map__zip2,axiom,
    ! [K: bool,L: list_nat] : map_na3335386563548176684ol_nat(produc8874003862241262736ol_nat(K),L) = zip_bool_nat(replicate_bool(aa_list_nat_nat(size_size_list_nat,L),K),L) ).

% map_zip2
tff(fact_6088_map__zip2,axiom,
    ! [K: nat,L: list_nat] : map_na7298421622053143531at_nat(product_Pair_nat_nat(K),L) = zip_nat_nat(replicate_nat(aa_list_nat_nat(size_size_list_nat,L),K),L) ).

% map_zip2
tff(fact_6089_map__zip2,axiom,
    ! [K: vEBT_VEBT,L: list_nat] : map_na4631810538828370761BT_nat(produc738532404422230701BT_nat(K),L) = zip_VEBT_VEBT_nat(aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_nat_nat(size_size_list_nat,L)),K),L) ).

% map_zip2
tff(fact_6090_map__zip2,axiom,
    ! [K: vEBT_VEBT,L: list_int] : map_in8151279748432256513BT_int(produc736041933913180425BT_int(K),L) = zip_VEBT_VEBT_int(aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_list_int_nat(size_size_list_int,L)),K),L) ).

% map_zip2
tff(fact_6091_image__fold__insert,axiom,
    ! [A2: set_nat,F: fun_nat_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( image_nat_VEBT_VEBT(F,A2) = finite7237651155313779962T_VEBT(aTP_Lamm_vp(F),bot_bo8194388402131092736T_VEBT,A2) ) ) ).

% image_fold_insert
tff(fact_6092_image__fold__insert,axiom,
    ! [A2: set_int,F: fun_int_VEBT_VEBT] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( image_int_VEBT_VEBT(F,A2) = finite7046945240081515934T_VEBT(aTP_Lamm_vq(F),bot_bo8194388402131092736T_VEBT,A2) ) ) ).

% image_fold_insert
tff(fact_6093_image__fold__insert,axiom,
    ! [A2: set_complex,F: fun_co4340422964966740447T_VEBT] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( image_932796090930683071T_VEBT(F,A2) = finite3784847964138203548T_VEBT(aTP_Lamm_vr(F),bot_bo8194388402131092736T_VEBT,A2) ) ) ).

% image_fold_insert
tff(fact_6094_image__fold__insert,axiom,
    ! [A2: set_Code_integer,F: fun_Co3766629010929971184T_VEBT] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( image_2664455355285265616T_VEBT(F,A2) = finite2306045077591485357T_VEBT(aTP_Lamm_vs(F),bot_bo8194388402131092736T_VEBT,A2) ) ) ).

% image_fold_insert
tff(fact_6095_image__fold__insert,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_set_real(image_nat_real(F),A2) = finite6086821452261492220t_real(aTP_Lamm_vt(F),bot_bot_set_real,A2) ) ) ).

% image_fold_insert
tff(fact_6096_image__fold__insert,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( image_int_real(F,A2) = finite2872259066402291288t_real(aTP_Lamm_vu(F),bot_bot_set_real,A2) ) ) ).

% image_fold_insert
tff(fact_6097_image__fold__insert,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( image_complex_real(F,A2) = finite2789990275036127706t_real(aTP_Lamm_vv(F),bot_bot_set_real,A2) ) ) ).

% image_fold_insert
tff(fact_6098_image__fold__insert,axiom,
    ! [A2: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( image_7738145705984076560r_real(F,A2) = finite7676789782520586249t_real(aTP_Lamm_vw(F),bot_bot_set_real,A2) ) ) ).

% image_fold_insert
tff(fact_6099_image__fold__insert,axiom,
    ! [A2: set_nat,F: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( image_nat_bool(F,A2) = finite186513945349748061t_bool(aTP_Lamm_vx(F),bot_bot_set_bool,A2) ) ) ).

% image_fold_insert
tff(fact_6100_image__fold__insert,axiom,
    ! [A2: set_int,F: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( image_int_bool(F,A2) = finite2916118514948564481t_bool(aTP_Lamm_vy(F),bot_bot_set_bool,A2) ) ) ).

% image_fold_insert
tff(fact_6101_fun__upd__image,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Y: nat] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_VEBT_VEBT_nat(fun_up6512855943550542919BT_nat(F,X3,Y),A2) = aa_set_nat_set_nat(insert_nat(Y),image_VEBT_VEBT_nat(F,minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)))) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_VEBT_VEBT_nat(fun_up6512855943550542919BT_nat(F,X3,Y),A2) = image_VEBT_VEBT_nat(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6102_fun__upd__image,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,Y: vEBT_VEBT] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_3375948659692109573T_VEBT(fun_up224749957652071293T_VEBT(F,X3,Y),A2) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Y),image_3375948659692109573T_VEBT(F,minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)))) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_3375948659692109573T_VEBT(fun_up224749957652071293T_VEBT(F,X3,Y),A2) = image_3375948659692109573T_VEBT(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6103_fun__upd__image,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,Y: int] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_VEBT_VEBT_int(fun_up6510365473041492643BT_int(F,X3,Y),A2) = aa_set_int_set_int(insert_int(Y),image_VEBT_VEBT_int(F,minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)))) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_VEBT_VEBT_int(fun_up6510365473041492643BT_int(F,X3,Y),A2) = image_VEBT_VEBT_int(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6104_fun__upd__image,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_bool,Y: bool] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_VEBT_VEBT_bool(fun_up7442040734458531168T_bool(F,X3,Y),A2) = aa_set_bool_set_bool(insert_bool(Y),image_VEBT_VEBT_bool(F,minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)))) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_VEBT_VEBT_bool(fun_up7442040734458531168T_bool(F,X3,Y),A2) = image_VEBT_VEBT_bool(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6105_fun__upd__image,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,Y: real] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_VEBT_VEBT_real(fun_up7749720967766031267T_real(F,X3,Y),A2) = aa_set_real_set_real(insert_real(Y),image_VEBT_VEBT_real(F,minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)))) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
       => ( image_VEBT_VEBT_real(fun_up7749720967766031267T_real(F,X3,Y),A2) = image_VEBT_VEBT_real(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6106_fun__upd__image,axiom,
    ! [X3: real,A2: set_real,F: fun_real_nat,Y: nat] :
      ( ( pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_nat(fun_upd_real_nat(F,X3,Y),A2) = aa_set_nat_set_nat(insert_nat(Y),image_real_nat(F,minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_nat(fun_upd_real_nat(F,X3,Y),A2) = image_real_nat(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6107_fun__upd__image,axiom,
    ! [X3: real,A2: set_real,F: fun_real_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_VEBT_VEBT(fun_up6563732700392937161T_VEBT(F,X3,Y),A2) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Y),image_real_VEBT_VEBT(F,minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_VEBT_VEBT(fun_up6563732700392937161T_VEBT(F,X3,Y),A2) = image_real_VEBT_VEBT(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6108_fun__upd__image,axiom,
    ! [X3: real,A2: set_real,F: fun_real_int,Y: int] :
      ( ( pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_int(fun_upd_real_int(F,X3,Y),A2) = aa_set_int_set_int(insert_int(Y),image_real_int(F,minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_int(fun_upd_real_int(F,X3,Y),A2) = image_real_int(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6109_fun__upd__image,axiom,
    ! [X3: real,A2: set_real,F: fun_real_bool,Y: bool] :
      ( ( pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_bool(fun_upd_real_bool(F,X3,Y),A2) = aa_set_bool_set_bool(insert_bool(Y),image_real_bool(F,minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_bool(fun_upd_real_bool(F,X3,Y),A2) = image_real_bool(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6110_fun__upd__image,axiom,
    ! [X3: real,A2: set_real,F: fun_real_real,Y: real] :
      ( ( pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_real(fun_upd_real_real(F,X3,Y),A2) = aa_set_real_set_real(insert_real(Y),image_real_real(F,minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),A2))
       => ( image_real_real(fun_upd_real_real(F,X3,Y),A2) = image_real_real(F,A2) ) ) ) ).

% fun_upd_image
tff(fact_6111_translation__subtract__diff,axiom,
    ! [A: real,S2: set_real,T: set_real] : image_real_real(aTP_Lamm_vz(A),minus_minus_set_real(S2,T)) = minus_minus_set_real(image_real_real(aTP_Lamm_vz(A),S2),image_real_real(aTP_Lamm_vz(A),T)) ).

% translation_subtract_diff
tff(fact_6112_translation__subtract__diff,axiom,
    ! [A: rat,S2: set_rat,T: set_rat] : image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_wa,A),minus_minus_set_rat(S2,T)) = minus_minus_set_rat(image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_wa,A),S2),image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_wa,A),T)) ).

% translation_subtract_diff
tff(fact_6113_translation__subtract__diff,axiom,
    ! [A: int,S2: set_int,T: set_int] : aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_wb,A)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S2),T)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_wb,A)),S2)),aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_wb,A)),T)) ).

% translation_subtract_diff
tff(fact_6114_restrict__map__upds,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,D4: set_VEBT_VEBT,M: fun_VE3810466045197099715n_real] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),D4))
       => ( restri7804378014469731632T_real(map_up4960461728960030954T_real(M,Xs,Ys),D4) = map_up4960461728960030954T_real(restri7804378014469731632T_real(M,minus_5127226145743854075T_VEBT(D4,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6115_restrict__map__upds,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_bool,D4: set_VEBT_VEBT,M: fun_VE6525960846995940460n_bool] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),D4))
       => ( restri6965680884402626707T_bool(map_up8421293420893609561T_bool(M,Xs,Ys),D4) = map_up8421293420893609561T_bool(restri6965680884402626707T_bool(M,minus_5127226145743854075T_VEBT(D4,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6116_restrict__map__upds,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,D4: set_VEBT_VEBT,M: fun_VE8174548813682830439on_nat] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),D4))
       => ( restri774867724463461460BT_nat(map_up889340208046081294BT_nat(M,Xs,Ys),D4) = map_up889340208046081294BT_nat(restri774867724463461460BT_nat(M,minus_5127226145743854075T_VEBT(D4,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6117_restrict__map__upds,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,D4: set_VEBT_VEBT,M: fun_VE5684465668751103683on_int] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),D4))
       => ( restri772377253954411184BT_int(map_up886849737537031018BT_int(M,Xs,Ys),D4) = map_up886849737537031018BT_int(restri772377253954411184BT_int(M,minus_5127226145743854075T_VEBT(D4,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6118_restrict__map__upds,axiom,
    ! [Xs: list_real,Ys: list_real,D4: set_real,M: fun_real_option_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),D4))
       => ( restri4420043737358336266l_real(map_upds_real_real(M,Xs,Ys),D4) = map_upds_real_real(restri4420043737358336266l_real(M,minus_minus_set_real(D4,aa_lis3769085826154824284t_real(set_real2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6119_restrict__map__upds,axiom,
    ! [Xs: list_real,Ys: list_bool,D4: set_real,M: fun_real_option_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),D4))
       => ( restri565833183434686009l_bool(map_upds_real_bool(M,Xs,Ys),D4) = map_upds_real_bool(restri565833183434686009l_bool(M,minus_minus_set_real(D4,aa_lis3769085826154824284t_real(set_real2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6120_restrict__map__upds,axiom,
    ! [Xs: list_real,Ys: list_nat,D4: set_real,M: fun_real_option_nat] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),D4))
       => ( restri6827137924477938990al_nat(map_upds_real_nat(M,Xs,Ys),D4) = map_upds_real_nat(restri6827137924477938990al_nat(M,minus_minus_set_real(D4,aa_lis3769085826154824284t_real(set_real2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6121_restrict__map__upds,axiom,
    ! [Xs: list_real,Ys: list_int,D4: set_real,M: fun_real_option_int] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),D4))
       => ( restri6824647453968888714al_int(map_upds_real_int(M,Xs,Ys),D4) = map_upds_real_int(restri6824647453968888714al_int(M,minus_minus_set_real(D4,aa_lis3769085826154824284t_real(set_real2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6122_restrict__map__upds,axiom,
    ! [Xs: list_bool,Ys: list_real,D4: set_bool,M: fun_bool_option_real] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs)),D4))
       => ( restri4802872615125879187l_real(map_upds_bool_real(M,Xs,Ys),D4) = map_upds_bool_real(restri4802872615125879187l_real(M,minus_minus_set_bool(D4,aa_lis3587183946281582166t_bool(set_bool2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6123_restrict__map__upds,axiom,
    ! [Xs: list_bool,Ys: list_bool,D4: set_bool,M: fun_bool_option_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs)),D4))
       => ( restri2720007268489790320l_bool(map_upds_bool_bool(M,Xs,Ys),D4) = map_upds_bool_bool(restri2720007268489790320l_bool(M,minus_minus_set_bool(D4,aa_lis3587183946281582166t_bool(set_bool2,Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6124_nth__image,axiom,
    ! [L: nat,Xs: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),size_s7982070591426661849_VEBTi(Xs)))
     => ( image_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),L)) = set_VEBT_VEBTi2(take_VEBT_VEBTi(L,Xs)) ) ) ).

% nth_image
tff(fact_6125_nth__image,axiom,
    ! [L: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),size_s3254054031482475050et_nat(Xs)))
     => ( aa_set7246812951530217472et_nat(image_nat_set_nat(nth_set_nat(Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),L)) = set_set_nat2(take_set_nat(L,Xs)) ) ) ).

% nth_image
tff(fact_6126_nth__image,axiom,
    ! [L: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),size_s6755466524823107622T_VEBT(Xs)))
     => ( image_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),L)) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(L,Xs)) ) ) ).

% nth_image
tff(fact_6127_nth__image,axiom,
    ! [L: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),aa_list_real_nat(size_size_list_real,Xs)))
     => ( aa_set_nat_set_real(image_nat_real(nth_real(Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),L)) = aa_lis3769085826154824284t_real(set_real2,take_real(L,Xs)) ) ) ).

% nth_image
tff(fact_6128_nth__image,axiom,
    ! [L: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( image_nat_bool(nth_bool(Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),L)) = aa_lis3587183946281582166t_bool(set_bool2,take_bool(L,Xs)) ) ) ).

% nth_image
tff(fact_6129_nth__image,axiom,
    ! [L: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),L)) = aa_list_nat_set_nat(set_nat2,take_nat(L,Xs)) ) ) ).

% nth_image
tff(fact_6130_nth__image,axiom,
    ! [L: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),aa_list_int_nat(size_size_list_int,Xs)))
     => ( aa_set_nat_set_int(image_nat_int(nth_int(Xs)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),L)) = aa_list_int_set_int(set_int2,take_int(L,Xs)) ) ) ).

% nth_image
tff(fact_6131_restrict__upd__same,axiom,
    ! [M: fun_VE8174548813682830439on_nat,X3: vEBT_VEBT,Y: nat] : restri774867724463461460BT_nat(fun_up5885881570350532375on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) = restri774867724463461460BT_nat(M,uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) ).

% restrict_upd_same
tff(fact_6132_restrict__upd__same,axiom,
    ! [M: fun_VE3889768403122177201on_num,X3: vEBT_VEBT,Y: num] : restri6555571547474015902BT_num(fun_up6594125507299370081on_num(M,X3,aa_num_option_num(some_num,Y)),uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) = restri6555571547474015902BT_num(M,uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) ).

% restrict_upd_same
tff(fact_6133_restrict__upd__same,axiom,
    ! [M: fun_real_option_nat,X3: real,Y: nat] : restri6827137924477938990al_nat(fun_up6677080212936659659on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),uminus612125837232591019t_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) = restri6827137924477938990al_nat(M,uminus612125837232591019t_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) ).

% restrict_upd_same
tff(fact_6134_restrict__upd__same,axiom,
    ! [M: fun_real_option_num,X3: real,Y: num] : restri3384469710633717624al_num(fun_up7385324149885497365on_num(M,X3,aa_num_option_num(some_num,Y)),uminus612125837232591019t_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) = restri3384469710633717624al_num(M,uminus612125837232591019t_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) ).

% restrict_upd_same
tff(fact_6135_restrict__upd__same,axiom,
    ! [M: fun_bool_option_nat,X3: bool,Y: nat] : restri5405821841207159351ol_nat(fun_up4506005684929255546on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) = restri5405821841207159351ol_nat(M,uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) ).

% restrict_upd_same
tff(fact_6136_restrict__upd__same,axiom,
    ! [M: fun_bool_option_num,X3: bool,Y: num] : restri1963153627362937985ol_num(fun_up5214249621878093252on_num(M,X3,aa_num_option_num(some_num,Y)),uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) = restri1963153627362937985ol_num(M,uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) ).

% restrict_upd_same
tff(fact_6137_restrict__upd__same,axiom,
    ! [M: fun_nat_option_nat,X3: nat,Y: nat] : restrict_map_nat_nat(fun_up1493157387958331631on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) = restrict_map_nat_nat(M,aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) ).

% restrict_upd_same
tff(fact_6138_restrict__upd__same,axiom,
    ! [M: fun_nat_option_num,X3: nat,Y: num] : restrict_map_nat_num(fun_up2201401324907169337on_num(M,X3,aa_num_option_num(some_num,Y)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) = restrict_map_nat_num(M,aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) ).

% restrict_upd_same
tff(fact_6139_restrict__upd__same,axiom,
    ! [M: fun_int_option_nat,X3: int,Y: nat] : restrict_map_int_nat(fun_up3620524117960394059on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) = restrict_map_int_nat(M,uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) ).

% restrict_upd_same
tff(fact_6140_restrict__upd__same,axiom,
    ! [M: fun_int_option_num,X3: int,Y: num] : restrict_map_int_num(fun_up4328768054909231765on_num(M,X3,aa_num_option_num(some_num,Y)),uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) = restrict_map_int_num(M,uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) ).

% restrict_upd_same
tff(fact_6141_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_6142_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_6143_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_6144_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_6145_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_6146_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_6147_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_6148_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_6149_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_6150_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_6151_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_6152_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_6153_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_6154_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_6155_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_6156_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_6157_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_6158_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_6159_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A) )
    <=> ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_6160_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_6161_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_6162_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_6163_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_6164_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_6165_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_6166_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_6167_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% neg_le_iff_le
tff(fact_6168_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% neg_le_iff_le
tff(fact_6169_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% neg_le_iff_le
tff(fact_6170_compl__le__compl__iff,axiom,
    ! [X3: assn,Y: assn] :
      ( ord_less_eq_assn(uminus_uminus_assn(X3),uminus_uminus_assn(Y))
    <=> ord_less_eq_assn(Y,X3) ) ).

% compl_le_compl_iff
tff(fact_6171_compl__le__compl__iff,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X3)),uminus1532241313380277803et_int(Y)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Y),X3)) ) ).

% compl_le_compl_iff
tff(fact_6172_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% neg_less_iff_less
tff(fact_6173_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% neg_less_iff_less
tff(fact_6174_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% neg_less_iff_less
tff(fact_6175_compl__less__compl__iff,axiom,
    ! [X3: assn,Y: assn] :
      ( ord_less_assn(uminus_uminus_assn(X3),uminus_uminus_assn(Y))
    <=> ord_less_assn(Y,X3) ) ).

% compl_less_compl_iff
tff(fact_6176_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_6177_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_6178_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_6179_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_6180_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)) = B ).

% minus_add_cancel
tff(fact_6181_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_6182_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).

% minus_add_cancel
tff(fact_6183_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_6184_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_6185_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_6186_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_6187_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_6188_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) ).

% mult_minus_right
tff(fact_6189_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_right
tff(fact_6190_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_right
tff(fact_6191_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_right
tff(fact_6192_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) ).

% minus_mult_minus
tff(fact_6193_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).

% minus_mult_minus
tff(fact_6194_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) ).

% minus_mult_minus
tff(fact_6195_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).

% minus_mult_minus
tff(fact_6196_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) ).

% mult_minus_left
tff(fact_6197_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_left
tff(fact_6198_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_left
tff(fact_6199_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_left
tff(fact_6200_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),A) ).

% minus_diff_eq
tff(fact_6201_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_6202_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% minus_diff_eq
tff(fact_6203_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_6204_abs__minus__cancel,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).

% abs_minus_cancel
tff(fact_6205_abs__minus__cancel,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus_cancel
tff(fact_6206_abs__minus__cancel,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus_cancel
tff(fact_6207_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_6208_abs__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) ).

% abs_minus
tff(fact_6209_abs__minus,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(abs_abs_complex,A) ).

% abs_minus
tff(fact_6210_abs__minus,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus
tff(fact_6211_abs__minus,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus
tff(fact_6212_abs__minus,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus
tff(fact_6213_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6214_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sgn_sgn_complex,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6215_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6216_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6217_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6218_Compl__anti__mono,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(B3)),uminus1532241313380277803et_int(A2))) ) ).

% Compl_anti_mono
tff(fact_6219_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(A2)),uminus1532241313380277803et_int(B3)))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2)) ) ).

% Compl_subset_Compl_iff
tff(fact_6220_image__map__upd,axiom,
    ! [X3: nat,A2: set_nat,M: fun_nat_option_nat,Y: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X3),A2))
     => ( image_nat_option_nat(fun_up1493157387958331631on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),A2) = image_nat_option_nat(M,A2) ) ) ).

% image_map_upd
tff(fact_6221_image__map__upd,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,M: fun_VE8174548813682830439on_nat,Y: nat] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
     => ( image_8844776943898047887on_nat(fun_up5885881570350532375on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),A2) = image_8844776943898047887on_nat(M,A2) ) ) ).

% image_map_upd
tff(fact_6222_image__map__upd,axiom,
    ! [X3: real,A2: set_real,M: fun_real_option_nat,Y: nat] :
      ( ~ pp(aa_set_real_bool(member_real(X3),A2))
     => ( image_5574646975249130067on_nat(fun_up6677080212936659659on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),A2) = image_5574646975249130067on_nat(M,A2) ) ) ).

% image_map_upd
tff(fact_6223_image__map__upd,axiom,
    ! [X3: int,A2: set_int,M: fun_int_option_nat,Y: nat] :
      ( ~ pp(aa_set_int_bool(member_int(X3),A2))
     => ( image_int_option_nat(fun_up3620524117960394059on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),A2) = image_int_option_nat(M,A2) ) ) ).

% image_map_upd
tff(fact_6224_image__map__upd,axiom,
    ! [X3: nat,A2: set_nat,M: fun_nat_option_num,Y: num] :
      ( ~ pp(aa_set_nat_bool(member_nat(X3),A2))
     => ( image_nat_option_num(fun_up2201401324907169337on_num(M,X3,aa_num_option_num(some_num,Y)),A2) = image_nat_option_num(M,A2) ) ) ).

% image_map_upd
tff(fact_6225_image__map__upd,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT,M: fun_VE3889768403122177201on_num,Y: num] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
     => ( image_329648843992109785on_num(fun_up6594125507299370081on_num(M,X3,aa_num_option_num(some_num,Y)),A2) = image_329648843992109785on_num(M,A2) ) ) ).

% image_map_upd
tff(fact_6226_image__map__upd,axiom,
    ! [X3: real,A2: set_real,M: fun_real_option_num,Y: num] :
      ( ~ pp(aa_set_real_bool(member_real(X3),A2))
     => ( image_6282890912197967773on_num(fun_up7385324149885497365on_num(M,X3,aa_num_option_num(some_num,Y)),A2) = image_6282890912197967773on_num(M,A2) ) ) ).

% image_map_upd
tff(fact_6227_image__map__upd,axiom,
    ! [X3: int,A2: set_int,M: fun_int_option_num,Y: num] :
      ( ~ pp(aa_set_int_bool(member_int(X3),A2))
     => ( image_int_option_num(fun_up4328768054909231765on_num(M,X3,aa_num_option_num(some_num,Y)),A2) = image_int_option_num(M,A2) ) ) ).

% image_map_upd
tff(fact_6228_image__map__upd,axiom,
    ! [X3: set_nat,A2: set_set_nat,M: fun_se8577451574693647267on_nat,Y: nat] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
     => ( image_2753855177019846445on_nat(fun_up5507366039599922085on_nat(M,X3,aa_nat_option_nat(some_nat,Y)),A2) = image_2753855177019846445on_nat(M,A2) ) ) ).

% image_map_upd
tff(fact_6229_image__map__upd,axiom,
    ! [X3: set_nat,A2: set_set_nat,M: fun_se4292671164132994029on_num,Y: num] :
      ( ~ pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
     => ( image_3462099113968684151on_num(fun_up6215609976548759791on_num(M,X3,aa_num_option_num(some_num,Y)),A2) = image_3462099113968684151on_num(M,A2) ) ) ).

% image_map_upd
tff(fact_6230_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_less_eq_nonneg
tff(fact_6231_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_less_eq_nonneg
tff(fact_6232_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_less_eq_nonneg
tff(fact_6233_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% less_eq_neg_nonpos
tff(fact_6234_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% less_eq_neg_nonpos
tff(fact_6235_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% less_eq_neg_nonpos
tff(fact_6236_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_le_0_iff_le
tff(fact_6237_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_le_0_iff_le
tff(fact_6238_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_le_0_iff_le
tff(fact_6239_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% neg_0_le_iff_le
tff(fact_6240_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% neg_0_le_iff_le
tff(fact_6241_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% neg_0_le_iff_le
tff(fact_6242_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_0_iff_less
tff(fact_6243_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_0_iff_less
tff(fact_6244_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_0_iff_less
tff(fact_6245_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% neg_0_less_iff_less
tff(fact_6246_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% neg_0_less_iff_less
tff(fact_6247_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% neg_0_less_iff_less
tff(fact_6248_neg__less__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_pos
tff(fact_6249_neg__less__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_pos
tff(fact_6250_neg__less__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_pos
tff(fact_6251_less__neg__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% less_neg_neg
tff(fact_6252_less__neg__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% less_neg_neg
tff(fact_6253_less__neg__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% less_neg_neg
tff(fact_6254_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_6255_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_6256_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_6257_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_6258_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_6259_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_6260_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_6261_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_6262_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_6263_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_6264_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_6265_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_6266_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_6267_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_6268_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_6269_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_6270_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_6271_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_6272_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_6273_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_6274_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_6275_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_6276_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_6277_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_6278_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) ).

% diff_minus_eq_add
tff(fact_6279_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_6280_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).

% diff_minus_eq_add
tff(fact_6281_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_6282_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),A) ).

% uminus_add_conv_diff
tff(fact_6283_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_6284_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% uminus_add_conv_diff
tff(fact_6285_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_6286_take__Suc__Cons,axiom,
    ! [N: nat,X3: bool,Xs: list_bool] : take_bool(aa_nat_nat(suc,N),cons_bool(X3,Xs)) = cons_bool(X3,take_bool(N,Xs)) ).

% take_Suc_Cons
tff(fact_6287_take__Suc__Cons,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] : take_nat(aa_nat_nat(suc,N),aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_list_nat_list_nat(cons_nat(X3),take_nat(N,Xs)) ).

% take_Suc_Cons
tff(fact_6288_take__Suc__Cons,axiom,
    ! [N: nat,X3: int,Xs: list_int] : take_int(aa_nat_nat(suc,N),aa_list_int_list_int(cons_int(X3),Xs)) = aa_list_int_list_int(cons_int(X3),take_int(N,Xs)) ).

% take_Suc_Cons
tff(fact_6289_take__all,axiom,
    ! [Xs: list_real,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),N))
     => ( take_real(N,Xs) = Xs ) ) ).

% take_all
tff(fact_6290_take__all,axiom,
    ! [Xs: list_bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),N))
     => ( take_bool(N,Xs) = Xs ) ) ).

% take_all
tff(fact_6291_take__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),N))
     => ( take_nat(N,Xs) = Xs ) ) ).

% take_all
tff(fact_6292_take__all,axiom,
    ! [Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),N))
     => ( take_int(N,Xs) = Xs ) ) ).

% take_all
tff(fact_6293_take__all__iff,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( take_real(N,Xs) = Xs )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),N)) ) ).

% take_all_iff
tff(fact_6294_take__all__iff,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( take_bool(N,Xs) = Xs )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),N)) ) ).

% take_all_iff
tff(fact_6295_take__all__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( take_nat(N,Xs) = Xs )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),N)) ) ).

% take_all_iff
tff(fact_6296_take__all__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( take_int(N,Xs) = Xs )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),N)) ) ).

% take_all_iff
tff(fact_6297_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(take_VEBT_VEBTi(N,Xs)),I) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I) ) ) ).

% nth_take
tff(fact_6298_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(take_VEBT_VEBT(N,Xs)),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I) ) ) ).

% nth_take
tff(fact_6299_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_int(nth_int(take_int(N,Xs)),I) = aa_nat_int(nth_int(Xs),I) ) ) ).

% nth_take
tff(fact_6300_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_nat(nth_nat(take_nat(N,Xs)),I) = aa_nat_nat(nth_nat(Xs),I) ) ) ).

% nth_take
tff(fact_6301_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(nth_bool(take_bool(N,Xs)),I))
      <=> pp(aa_nat_bool(nth_bool(Xs),I)) ) ) ).

% nth_take
tff(fact_6302_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( take_nat(N,list_update_nat(Xs,M,Y)) = take_nat(N,Xs) ) ) ).

% take_update_cancel
tff(fact_6303_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_bool,Y: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( take_bool(N,list_update_bool(Xs,M,Y)) = take_bool(N,Xs) ) ) ).

% take_update_cancel
tff(fact_6304_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( take_VEBT_VEBTi(N,list_u6098035379799741383_VEBTi(Xs,M,Y)) = take_VEBT_VEBTi(N,Xs) ) ) ).

% take_update_cancel
tff(fact_6305_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( take_VEBT_VEBT(N,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,M),Y)) = take_VEBT_VEBT(N,Xs) ) ) ).

% take_update_cancel
tff(fact_6306_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_6307_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_6308_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_6309_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_6310_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_6311_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_6312_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_6313_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_6314_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_6315_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_6316_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_6317_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_6318_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_6319_abs__of__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_6320_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_6321_abs__of__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_6322_subset__Compl__singleton,axiom,
    ! [A2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT))))
    <=> ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_6323_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(aa_set1013257548857649334et_nat(insert_set_nat(B),bot_bot_set_set_nat))))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_6324_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),uminus612125837232591019t_real(aa_set_real_set_real(insert_real(B),bot_bot_set_real))))
    <=> ~ pp(aa_set_real_bool(member_real(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_6325_subset__Compl__singleton,axiom,
    ! [A2: set_bool,B: bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(B),bot_bot_set_bool))))
    <=> ~ pp(aa_set_bool_bool(member_bool(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_6326_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat))))
    <=> ~ pp(aa_set_nat_bool(member_nat(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_6327_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(B),bot_bot_set_int))))
    <=> ~ pp(aa_set_int_bool(member_int(B),A2)) ) ).

% subset_Compl_singleton
tff(fact_6328_map__upds__list__update2__drop,axiom,
    ! [Xs: list_real,I: nat,M: fun_re869386379898482478_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),I))
     => ( map_up4277394084060657369_VEBTi(M,Xs,list_u6098035379799741383_VEBTi(Ys,I,Y)) = map_up4277394084060657369_VEBTi(M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6329_map__upds__list__update2__drop,axiom,
    ! [Xs: list_real,I: nat,M: fun_re6149904099307231015T_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),I))
     => ( map_up3774473461586936848T_VEBT(M,Xs,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Ys,I),Y)) = map_up3774473461586936848T_VEBT(M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6330_map__upds__list__update2__drop,axiom,
    ! [Xs: list_bool,I: nat,M: fun_bo5052482999933817007_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),I))
     => ( map_up3574140571279353302_VEBTi(M,Xs,list_u6098035379799741383_VEBTi(Ys,I,Y)) = map_up3574140571279353302_VEBTi(M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6331_map__upds__list__update2__drop,axiom,
    ! [Xs: list_bool,I: nat,M: fun_bo8890943138197189862T_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),I))
     => ( map_up5237471190051075923T_VEBT(M,Xs,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Ys,I),Y)) = map_up5237471190051075923T_VEBT(M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6332_map__upds__list__update2__drop,axiom,
    ! [Xs: list_nat,I: nat,M: fun_na7681084296160214226_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),I))
     => ( map_up547622734759332733_VEBTi(M,Xs,list_u6098035379799741383_VEBTi(Ys,I,Y)) = map_up547622734759332733_VEBTi(M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6333_map__upds__list__update2__drop,axiom,
    ! [Xs: list_nat,I: nat,M: fun_na2183005016389635715T_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),I))
     => ( map_up750602437722059884T_VEBT(M,Xs,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Ys,I),Y)) = map_up750602437722059884T_VEBT(M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6334_map__upds__list__update2__drop,axiom,
    ! [Xs: list_int,I: nat,M: fun_in2850175258336333102_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),I))
     => ( map_up2395700066158073305_VEBTi(M,Xs,list_u6098035379799741383_VEBTi(Ys,I,Y)) = map_up2395700066158073305_VEBTi(M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6335_map__upds__list__update2__drop,axiom,
    ! [Xs: list_int,I: nat,M: fun_in6246267328955250983T_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),I))
     => ( map_up3480207007320876304T_VEBT(M,Xs,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Ys,I),Y)) = map_up3480207007320876304T_VEBT(M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6336_map__upds__Cons,axiom,
    ! [M: fun_nat_option_int,A: nat,As: list_nat,B: int,Bs: list_int] : map_upds_nat_int(M,aa_list_nat_list_nat(cons_nat(A),As),aa_list_int_list_int(cons_int(B),Bs)) = map_upds_nat_int(fun_up6538678405303910731on_int(M,A,aa_int_option_int(some_int,B)),As,Bs) ).

% map_upds_Cons
tff(fact_6337_map__upds__Cons,axiom,
    ! [M: fun_int_option_int,A: int,As: list_int,B: int,Bs: list_int] : map_upds_int_int(M,aa_list_int_list_int(cons_int(A),As),aa_list_int_list_int(cons_int(B),Bs)) = map_upds_int_int(fun_up8666045135305973159on_int(M,A,aa_int_option_int(some_int,B)),As,Bs) ).

% map_upds_Cons
tff(fact_6338_map__upds__Cons,axiom,
    ! [M: fun_na7803622437922444752at_nat,A: nat,As: list_nat,B: product_prod_nat_nat,Bs: list_P6011104703257516679at_nat] : map_up1590851270032768825at_nat(M,aa_list_nat_list_nat(cons_nat(A),As),cons_P6512896166579812791at_nat(B,Bs)) = map_up1590851270032768825at_nat(fun_up2252230070437638584at_nat(M,A,aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B)),As,Bs) ).

% map_upds_Cons
tff(fact_6339_map__upds__Cons,axiom,
    ! [M: fun_in7291707798943782516at_nat,A: int,As: list_int,B: product_prod_nat_nat,Bs: list_P6011104703257516679at_nat] : map_up4628693354496902621at_nat(M,aa_list_int_list_int(cons_int(A),As),cons_P6512896166579812791at_nat(B,Bs)) = map_up4628693354496902621at_nat(fun_up1740315431458976348at_nat(M,A,aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,B)),As,Bs) ).

% map_upds_Cons
tff(fact_6340_map__upds__Cons,axiom,
    ! [M: fun_nat_option_nat,A: nat,As: list_nat,B: nat,Bs: list_nat] : map_upds_nat_nat(M,aa_list_nat_list_nat(cons_nat(A),As),aa_list_nat_list_nat(cons_nat(B),Bs)) = map_upds_nat_nat(fun_up1493157387958331631on_nat(M,A,aa_nat_option_nat(some_nat,B)),As,Bs) ).

% map_upds_Cons
tff(fact_6341_map__upds__Cons,axiom,
    ! [M: fun_int_option_nat,A: int,As: list_int,B: nat,Bs: list_nat] : map_upds_int_nat(M,aa_list_int_list_int(cons_int(A),As),aa_list_nat_list_nat(cons_nat(B),Bs)) = map_upds_int_nat(fun_up3620524117960394059on_nat(M,A,aa_nat_option_nat(some_nat,B)),As,Bs) ).

% map_upds_Cons
tff(fact_6342_map__upds__Cons,axiom,
    ! [M: fun_nat_option_num,A: nat,As: list_nat,B: num,Bs: list_num] : map_upds_nat_num(M,aa_list_nat_list_nat(cons_nat(A),As),cons_num(B,Bs)) = map_upds_nat_num(fun_up2201401324907169337on_num(M,A,aa_num_option_num(some_num,B)),As,Bs) ).

% map_upds_Cons
tff(fact_6343_map__upds__Cons,axiom,
    ! [M: fun_int_option_num,A: int,As: list_int,B: num,Bs: list_num] : map_upds_int_num(M,aa_list_int_list_int(cons_int(A),As),cons_num(B,Bs)) = map_upds_int_num(fun_up4328768054909231765on_num(M,A,aa_num_option_num(some_num,B)),As,Bs) ).

% map_upds_Cons
tff(fact_6344_map__upds__twist,axiom,
    ! [A: int,As: list_int,M: fun_int_option_nat,B: nat,Bs: list_nat] :
      ( ~ pp(aa_set_int_bool(member_int(A),aa_list_int_set_int(set_int2,As)))
     => ( map_upds_int_nat(fun_up3620524117960394059on_nat(M,A,aa_nat_option_nat(some_nat,B)),As,Bs) = fun_up3620524117960394059on_nat(map_upds_int_nat(M,As,Bs),A,aa_nat_option_nat(some_nat,B)) ) ) ).

% map_upds_twist
tff(fact_6345_map__upds__twist,axiom,
    ! [A: vEBT_VEBT,As: list_VEBT_VEBT,M: fun_VE8174548813682830439on_nat,B: nat,Bs: list_nat] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,As)))
     => ( map_up889340208046081294BT_nat(fun_up5885881570350532375on_nat(M,A,aa_nat_option_nat(some_nat,B)),As,Bs) = fun_up5885881570350532375on_nat(map_up889340208046081294BT_nat(M,As,Bs),A,aa_nat_option_nat(some_nat,B)) ) ) ).

% map_upds_twist
tff(fact_6346_map__upds__twist,axiom,
    ! [A: nat,As: list_nat,M: fun_nat_option_nat,B: nat,Bs: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(A),aa_list_nat_set_nat(set_nat2,As)))
     => ( map_upds_nat_nat(fun_up1493157387958331631on_nat(M,A,aa_nat_option_nat(some_nat,B)),As,Bs) = fun_up1493157387958331631on_nat(map_upds_nat_nat(M,As,Bs),A,aa_nat_option_nat(some_nat,B)) ) ) ).

% map_upds_twist
tff(fact_6347_map__upds__twist,axiom,
    ! [A: real,As: list_real,M: fun_real_option_nat,B: nat,Bs: list_nat] :
      ( ~ pp(aa_set_real_bool(member_real(A),aa_lis3769085826154824284t_real(set_real2,As)))
     => ( map_upds_real_nat(fun_up6677080212936659659on_nat(M,A,aa_nat_option_nat(some_nat,B)),As,Bs) = fun_up6677080212936659659on_nat(map_upds_real_nat(M,As,Bs),A,aa_nat_option_nat(some_nat,B)) ) ) ).

% map_upds_twist
tff(fact_6348_map__upds__twist,axiom,
    ! [A: bool,As: list_bool,M: fun_bool_option_nat,B: nat,Bs: list_nat] :
      ( ~ pp(aa_set_bool_bool(member_bool(A),aa_lis3587183946281582166t_bool(set_bool2,As)))
     => ( map_upds_bool_nat(fun_up4506005684929255546on_nat(M,A,aa_nat_option_nat(some_nat,B)),As,Bs) = fun_up4506005684929255546on_nat(map_upds_bool_nat(M,As,Bs),A,aa_nat_option_nat(some_nat,B)) ) ) ).

% map_upds_twist
tff(fact_6349_map__upds__twist,axiom,
    ! [A: int,As: list_int,M: fun_int_option_num,B: num,Bs: list_num] :
      ( ~ pp(aa_set_int_bool(member_int(A),aa_list_int_set_int(set_int2,As)))
     => ( map_upds_int_num(fun_up4328768054909231765on_num(M,A,aa_num_option_num(some_num,B)),As,Bs) = fun_up4328768054909231765on_num(map_upds_int_num(M,As,Bs),A,aa_num_option_num(some_num,B)) ) ) ).

% map_upds_twist
tff(fact_6350_map__upds__twist,axiom,
    ! [A: vEBT_VEBT,As: list_VEBT_VEBT,M: fun_VE3889768403122177201on_num,B: num,Bs: list_num] :
      ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,As)))
     => ( map_up6670044031056635736BT_num(fun_up6594125507299370081on_num(M,A,aa_num_option_num(some_num,B)),As,Bs) = fun_up6594125507299370081on_num(map_up6670044031056635736BT_num(M,As,Bs),A,aa_num_option_num(some_num,B)) ) ) ).

% map_upds_twist
tff(fact_6351_map__upds__twist,axiom,
    ! [A: nat,As: list_nat,M: fun_nat_option_num,B: num,Bs: list_num] :
      ( ~ pp(aa_set_nat_bool(member_nat(A),aa_list_nat_set_nat(set_nat2,As)))
     => ( map_upds_nat_num(fun_up2201401324907169337on_num(M,A,aa_num_option_num(some_num,B)),As,Bs) = fun_up2201401324907169337on_num(map_upds_nat_num(M,As,Bs),A,aa_num_option_num(some_num,B)) ) ) ).

% map_upds_twist
tff(fact_6352_map__upds__twist,axiom,
    ! [A: real,As: list_real,M: fun_real_option_num,B: num,Bs: list_num] :
      ( ~ pp(aa_set_real_bool(member_real(A),aa_lis3769085826154824284t_real(set_real2,As)))
     => ( map_upds_real_num(fun_up7385324149885497365on_num(M,A,aa_num_option_num(some_num,B)),As,Bs) = fun_up7385324149885497365on_num(map_upds_real_num(M,As,Bs),A,aa_num_option_num(some_num,B)) ) ) ).

% map_upds_twist
tff(fact_6353_map__upds__twist,axiom,
    ! [A: bool,As: list_bool,M: fun_bool_option_num,B: num,Bs: list_num] :
      ( ~ pp(aa_set_bool_bool(member_bool(A),aa_lis3587183946281582166t_bool(set_bool2,As)))
     => ( map_upds_bool_num(fun_up5214249621878093252on_num(M,A,aa_num_option_num(some_num,B)),As,Bs) = fun_up5214249621878093252on_num(map_upds_bool_num(M,As,Bs),A,aa_num_option_num(some_num,B)) ) ) ).

% map_upds_twist
tff(fact_6354_sgn__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_6355_sgn__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_6356_sgn__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_6357_sgn__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_6358_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = A ) ) ).

% minus_equation_iff
tff(fact_6359_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_6360_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_6361_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_6362_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_6363_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_6364_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_6365_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_6366_Collect__neg__eq,axiom,
    ! [P: fun_complex_bool] : collect_complex(aTP_Lamm_wc(P)) = uminus8566677241136511917omplex(collect_complex(P)) ).

% Collect_neg_eq
tff(fact_6367_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_bool] : collect_list_nat(aTP_Lamm_wd(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_6368_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_bool] : collect_set_nat(aTP_Lamm_we(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_6369_Collect__neg__eq,axiom,
    ! [P: fun_nat_bool] : collect_nat(aTP_Lamm_wf(P)) = aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P)) ).

% Collect_neg_eq
tff(fact_6370_Collect__neg__eq,axiom,
    ! [P: fun_int_bool] : collect_int(aTP_Lamm_wg(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_6371_Compl__eq,axiom,
    ! [A2: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(A2) = collect_VEBT_VEBT(aTP_Lamm_wh(A2)) ).

% Compl_eq
tff(fact_6372_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_wi(A2)) ).

% Compl_eq
tff(fact_6373_Compl__eq,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(aTP_Lamm_wj(A2)) ).

% Compl_eq
tff(fact_6374_Compl__eq,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(aTP_Lamm_wk(A2)) ).

% Compl_eq
tff(fact_6375_Compl__eq,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_wl(A2)) ).

% Compl_eq
tff(fact_6376_Compl__eq,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2) = collect_nat(aTP_Lamm_wm(A2)) ).

% Compl_eq
tff(fact_6377_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_wn(A2)) ).

% Compl_eq
tff(fact_6378_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% le_imp_neg_le
tff(fact_6379_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_imp_neg_le
tff(fact_6380_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% le_imp_neg_le
tff(fact_6381_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_le_iff
tff(fact_6382_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_le_iff
tff(fact_6383_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_le_iff
tff(fact_6384_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% le_minus_iff
tff(fact_6385_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_minus_iff
tff(fact_6386_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% le_minus_iff
tff(fact_6387_compl__mono,axiom,
    ! [X3: assn,Y: assn] :
      ( ord_less_eq_assn(X3,Y)
     => ord_less_eq_assn(uminus_uminus_assn(Y),uminus_uminus_assn(X3)) ) ).

% compl_mono
tff(fact_6388_compl__mono,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(X3))) ) ).

% compl_mono
tff(fact_6389_compl__le__swap1,axiom,
    ! [Y: assn,X3: assn] :
      ( ord_less_eq_assn(Y,uminus_uminus_assn(X3))
     => ord_less_eq_assn(X3,uminus_uminus_assn(Y)) ) ).

% compl_le_swap1
tff(fact_6390_compl__le__swap1,axiom,
    ! [Y: set_int,X3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),uminus1532241313380277803et_int(X3)))
     => pp(aa_set_int_bool(ord_less_eq_set_int(X3),uminus1532241313380277803et_int(Y))) ) ).

% compl_le_swap1
tff(fact_6391_compl__le__swap2,axiom,
    ! [Y: assn,X3: assn] :
      ( ord_less_eq_assn(uminus_uminus_assn(Y),X3)
     => ord_less_eq_assn(uminus_uminus_assn(X3),Y) ) ).

% compl_le_swap2
tff(fact_6392_compl__le__swap2,axiom,
    ! [Y: set_int,X3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),X3))
     => pp(aa_set_int_bool(ord_less_eq_set_int(uminus1532241313380277803et_int(X3)),Y)) ) ).

% compl_le_swap2
tff(fact_6393_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_6394_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_6395_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_6396_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_less_iff
tff(fact_6397_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_less_iff
tff(fact_6398_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_less_iff
tff(fact_6399_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% less_minus_iff
tff(fact_6400_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% less_minus_iff
tff(fact_6401_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% less_minus_iff
tff(fact_6402_compl__less__swap1,axiom,
    ! [Y: assn,X3: assn] :
      ( ord_less_assn(Y,uminus_uminus_assn(X3))
     => ord_less_assn(X3,uminus_uminus_assn(Y)) ) ).

% compl_less_swap1
tff(fact_6403_compl__less__swap2,axiom,
    ! [Y: assn,X3: assn] :
      ( ord_less_assn(uminus_uminus_assn(Y),X3)
     => ord_less_assn(uminus_uminus_assn(X3),Y) ) ).

% compl_less_swap2
tff(fact_6404_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_6405_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_6406_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_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_6407_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_6408_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_6409_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_6410_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_6411_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = 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_6412_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_6413_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_6414_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_6415_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_6416_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),B) )
    <=> ( ( A = B )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_6417_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_6418_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_6419_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_6420_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_6421_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_6422_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_6423_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_6424_abs__eq__iff,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X3) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y) )
    <=> ( ( X3 = Y )
        | ( X3 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6425_abs__eq__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,X3) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( X3 = Y )
        | ( X3 = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6426_abs__eq__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( aa_rat_rat(abs_abs_rat,X3) = aa_rat_rat(abs_abs_rat,Y) )
    <=> ( ( X3 = Y )
        | ( X3 = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6427_abs__eq__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X3) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X3 = Y )
        | ( X3 = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6428_Collect__imp__eq,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] : collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_wo(P),Q)) = sup_sup_set_complex(uminus8566677241136511917omplex(collect_complex(P)),collect_complex(Q)) ).

% Collect_imp_eq
tff(fact_6429_Collect__imp__eq,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] : collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_wp(P),Q)) = sup_sup_set_list_nat(uminus3195874150345416415st_nat(collect_list_nat(P)),collect_list_nat(Q)) ).

% Collect_imp_eq
tff(fact_6430_Collect__imp__eq,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] : collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_wq(P),Q)) = sup_sup_set_set_nat(uminus613421341184616069et_nat(collect_set_nat(P)),collect_set_nat(Q)) ).

% Collect_imp_eq
tff(fact_6431_Collect__imp__eq,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] : collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_wr(P),Q)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(collect_int(P))),collect_int(Q)) ).

% Collect_imp_eq
tff(fact_6432_Collect__imp__eq,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] : collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_ws(P),Q)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P))),collect_nat(Q)) ).

% Collect_imp_eq
tff(fact_6433_translation__subtract__Compl,axiom,
    ! [A: real,T: set_real] : image_real_real(aTP_Lamm_vz(A),uminus612125837232591019t_real(T)) = uminus612125837232591019t_real(image_real_real(aTP_Lamm_vz(A),T)) ).

% translation_subtract_Compl
tff(fact_6434_translation__subtract__Compl,axiom,
    ! [A: rat,T: set_rat] : image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_wa,A),uminus2201863774496077783et_rat(T)) = uminus2201863774496077783et_rat(image_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_wa,A),T)) ).

% translation_subtract_Compl
tff(fact_6435_translation__subtract__Compl,axiom,
    ! [A: int,T: set_int] : aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_wb,A)),uminus1532241313380277803et_int(T)) = uminus1532241313380277803et_int(aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_wb,A)),T)) ).

% translation_subtract_Compl
tff(fact_6436_set__take__subset,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(N,Xs))),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))) ).

% set_take_subset
tff(fact_6437_set__take__subset,axiom,
    ! [N: nat,Xs: list_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,take_nat(N,Xs))),aa_list_nat_set_nat(set_nat2,Xs))) ).

% set_take_subset
tff(fact_6438_set__take__subset,axiom,
    ! [N: nat,Xs: list_real] : pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,take_real(N,Xs))),aa_lis3769085826154824284t_real(set_real2,Xs))) ).

% set_take_subset
tff(fact_6439_set__take__subset,axiom,
    ! [N: nat,Xs: list_bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,take_bool(N,Xs))),aa_lis3587183946281582166t_bool(set_bool2,Xs))) ).

% set_take_subset
tff(fact_6440_set__take__subset,axiom,
    ! [N: nat,Xs: list_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,take_int(N,Xs))),aa_list_int_set_int(set_int2,Xs))) ).

% set_take_subset
tff(fact_6441_sorted__take,axiom,
    ! [Xs: list_bool,N: nat] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
     => pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),take_bool(N,Xs))) ) ).

% sorted_take
tff(fact_6442_sorted__take,axiom,
    ! [Xs: list_rat,N: nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => sorted_wrt_rat(ord_less_eq_rat,take_rat(N,Xs)) ) ).

% sorted_take
tff(fact_6443_sorted__take,axiom,
    ! [Xs: list_num,N: nat] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => sorted_wrt_num(ord_less_eq_num,take_num(N,Xs)) ) ).

% sorted_take
tff(fact_6444_sorted__take,axiom,
    ! [Xs: list_nat,N: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),take_nat(N,Xs))) ) ).

% sorted_take
tff(fact_6445_sorted__take,axiom,
    ! [Xs: list_int,N: nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),take_int(N,Xs))) ) ).

% sorted_take
tff(fact_6446_None__notin__image__Some,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ~ member3954567711264315760at_nat(none_P5556105721700978146at_nat,image_4198897800814241419at_nat(some_P7363390416028606310at_nat,A2)) ).

% None_notin_image_Some
tff(fact_6447_None__notin__image__Some,axiom,
    ! [A2: set_nat] : ~ member_option_nat(none_nat,image_nat_option_nat(some_nat,A2)) ).

% None_notin_image_Some
tff(fact_6448_None__notin__image__Some,axiom,
    ! [A2: set_num] : ~ member_option_num(none_num,image_num_option_num(some_num,A2)) ).

% None_notin_image_Some
tff(fact_6449_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_6450_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_6451_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_6452_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_6453_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_6454_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_6455_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_6456_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_6457_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_6458_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_6459_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_6460_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_6461_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6462_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_6463_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6464_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_6465_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6466_neg__eq__iff__add__eq__0,axiom,
    ! [A: 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_6467_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6468_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_6469_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_6470_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_6471_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_6472_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_6473_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(4)
tff(fact_6474_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(4)
tff(fact_6475_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(4)
tff(fact_6476_le__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% le_minus_one_simps(2)
tff(fact_6477_le__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% le_minus_one_simps(2)
tff(fact_6478_le__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% le_minus_one_simps(2)
tff(fact_6479_less__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% less_minus_one_simps(2)
tff(fact_6480_less__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% less_minus_one_simps(2)
tff(fact_6481_less__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% less_minus_one_simps(2)
tff(fact_6482_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(4)
tff(fact_6483_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(4)
tff(fact_6484_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(4)
tff(fact_6485_square__eq__1__iff,axiom,
    ! [X3: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),X3) = one_one_complex )
    <=> ( ( X3 = one_one_complex )
        | ( X3 = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_6486_square__eq__1__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3) = one_one_real )
    <=> ( ( X3 = one_one_real )
        | ( X3 = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_6487_square__eq__1__iff,axiom,
    ! [X3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),X3) = one_one_rat )
    <=> ( ( X3 = one_one_rat )
        | ( X3 = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_6488_square__eq__1__iff,axiom,
    ! [X3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X3),X3) = one_one_int )
    <=> ( ( X3 = one_one_int )
        | ( X3 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_6489_group__cancel_Osub2,axiom,
    ! [B3: complex,K: complex,B: complex,A: complex] :
      ( ( B3 = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,K),B) )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B3) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6490_group__cancel_Osub2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(minus_minus_real(A),B3) = 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_6491_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6492_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3 = 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),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6493_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_6494_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_6495_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_6496_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_6497_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6498_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_6499_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6500_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_6501_map__upd__upds__conv__if,axiom,
    ! [X3: int,Ys: list_num,Xs: list_int,F: fun_int_option_num,Y: num] :
      ( ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,take_int(size_size_list_num(Ys),Xs))))
       => ( map_upds_int_num(fun_up4328768054909231765on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = map_upds_int_num(F,Xs,Ys) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,take_int(size_size_list_num(Ys),Xs))))
       => ( map_upds_int_num(fun_up4328768054909231765on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = fun_up4328768054909231765on_num(map_upds_int_num(F,Xs,Ys),X3,aa_num_option_num(some_num,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6502_map__upd__upds__conv__if,axiom,
    ! [X3: vEBT_VEBT,Ys: list_num,Xs: list_VEBT_VEBT,F: fun_VE3889768403122177201on_num,Y: num] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(size_size_list_num(Ys),Xs))))
       => ( map_up6670044031056635736BT_num(fun_up6594125507299370081on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = map_up6670044031056635736BT_num(F,Xs,Ys) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(size_size_list_num(Ys),Xs))))
       => ( map_up6670044031056635736BT_num(fun_up6594125507299370081on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = fun_up6594125507299370081on_num(map_up6670044031056635736BT_num(F,Xs,Ys),X3,aa_num_option_num(some_num,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6503_map__upd__upds__conv__if,axiom,
    ! [X3: nat,Ys: list_num,Xs: list_nat,F: fun_nat_option_num,Y: num] :
      ( ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,take_nat(size_size_list_num(Ys),Xs))))
       => ( map_upds_nat_num(fun_up2201401324907169337on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = map_upds_nat_num(F,Xs,Ys) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,take_nat(size_size_list_num(Ys),Xs))))
       => ( map_upds_nat_num(fun_up2201401324907169337on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = fun_up2201401324907169337on_num(map_upds_nat_num(F,Xs,Ys),X3,aa_num_option_num(some_num,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6504_map__upd__upds__conv__if,axiom,
    ! [X3: real,Ys: list_num,Xs: list_real,F: fun_real_option_num,Y: num] :
      ( ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,take_real(size_size_list_num(Ys),Xs))))
       => ( map_upds_real_num(fun_up7385324149885497365on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = map_upds_real_num(F,Xs,Ys) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,take_real(size_size_list_num(Ys),Xs))))
       => ( map_upds_real_num(fun_up7385324149885497365on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = fun_up7385324149885497365on_num(map_upds_real_num(F,Xs,Ys),X3,aa_num_option_num(some_num,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6505_map__upd__upds__conv__if,axiom,
    ! [X3: bool,Ys: list_num,Xs: list_bool,F: fun_bool_option_num,Y: num] :
      ( ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,take_bool(size_size_list_num(Ys),Xs))))
       => ( map_upds_bool_num(fun_up5214249621878093252on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = map_upds_bool_num(F,Xs,Ys) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,take_bool(size_size_list_num(Ys),Xs))))
       => ( map_upds_bool_num(fun_up5214249621878093252on_num(F,X3,aa_num_option_num(some_num,Y)),Xs,Ys) = fun_up5214249621878093252on_num(map_upds_bool_num(F,Xs,Ys),X3,aa_num_option_num(some_num,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6506_map__upd__upds__conv__if,axiom,
    ! [X3: int,Ys: list_real,Xs: list_int,F: fun_int_option_real,Y: real] :
      ( ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,take_int(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_upds_int_real(fun_up3065300406613234215n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = map_upds_int_real(F,Xs,Ys) ) )
      & ( ~ pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,take_int(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_upds_int_real(fun_up3065300406613234215n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = fun_up3065300406613234215n_real(map_upds_int_real(F,Xs,Ys),X3,aa_real_option_real(some_real,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6507_map__upd__upds__conv__if,axiom,
    ! [X3: vEBT_VEBT,Ys: list_real,Xs: list_VEBT_VEBT,F: fun_VE3810466045197099715n_real,Y: real] :
      ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_up4960461728960030954T_real(fun_up5420679954143522291n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = map_up4960461728960030954T_real(F,Xs,Ys) ) )
      & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_up4960461728960030954T_real(fun_up5420679954143522291n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = fun_up5420679954143522291n_real(map_up4960461728960030954T_real(F,Xs,Ys),X3,aa_real_option_real(some_real,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6508_map__upd__upds__conv__if,axiom,
    ! [X3: nat,Ys: list_real,Xs: list_nat,F: fun_nat_option_real,Y: real] :
      ( ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,take_nat(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_upds_nat_real(fun_up6877590261764547531n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = map_upds_nat_real(F,Xs,Ys) ) )
      & ( ~ pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,take_nat(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_upds_nat_real(fun_up6877590261764547531n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = fun_up6877590261764547531n_real(map_upds_nat_real(F,Xs,Ys),X3,aa_real_option_real(some_real,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6509_map__upd__upds__conv__if,axiom,
    ! [X3: real,Ys: list_real,Xs: list_real,F: fun_real_option_real,Y: real] :
      ( ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,take_real(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_upds_real_real(fun_up7779911115344551847n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = map_upds_real_real(F,Xs,Ys) ) )
      & ( ~ pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,take_real(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_upds_real_real(fun_up7779911115344551847n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = fun_up7779911115344551847n_real(map_upds_real_real(F,Xs,Ys),X3,aa_real_option_real(some_real,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6510_map__upd__upds__conv__if,axiom,
    ! [X3: bool,Ys: list_real,Xs: list_bool,F: fun_bool_option_real,Y: real] :
      ( ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,take_bool(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_upds_bool_real(fun_up723089428184825302n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = map_upds_bool_real(F,Xs,Ys) ) )
      & ( ~ pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,take_bool(aa_list_real_nat(size_size_list_real,Ys),Xs))))
       => ( map_upds_bool_real(fun_up723089428184825302n_real(F,X3,aa_real_option_real(some_real,Y)),Xs,Ys) = fun_up723089428184825302n_real(map_upds_bool_real(F,Xs,Ys),X3,aa_real_option_real(some_real,Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6511_abs__ge__minus__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))) ).

% abs_ge_minus_self
tff(fact_6512_abs__ge__minus__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A))) ).

% abs_ge_minus_self
tff(fact_6513_abs__ge__minus__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_minus_self
tff(fact_6514_abs__ge__minus__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A))) ).

% abs_ge_minus_self
tff(fact_6515_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_le_iff
tff(fact_6516_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_le_iff
tff(fact_6517_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_le_iff
tff(fact_6518_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_le_iff
tff(fact_6519_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ).

% abs_le_D2
tff(fact_6520_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ).

% abs_le_D2
tff(fact_6521_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ).

% abs_le_D2
tff(fact_6522_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ).

% abs_le_D2
tff(fact_6523_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B)) ) ) ).

% abs_leI
tff(fact_6524_abs__leI,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)) ) ) ).

% abs_leI
tff(fact_6525_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)) ) ) ).

% abs_leI
tff(fact_6526_abs__leI,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)) ) ) ).

% abs_leI
tff(fact_6527_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_less_iff
tff(fact_6528_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_less_iff
tff(fact_6529_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_less_iff
tff(fact_6530_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_less_iff
tff(fact_6531_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),uminus612125837232591019t_real(A2)))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_6532_subset__Compl__self__eq,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),uminus5166143097222041326t_bool(A2)))
    <=> ( A2 = bot_bot_set_bool ) ) ).

% subset_Compl_self_eq
tff(fact_6533_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_6534_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(A2)))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_6535_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) != zero_z3403309356797280102nteger )
       => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) != zero_z3403309356797280102nteger )
         => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6536_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6537_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) != aa_rat_rat(sgn_sgn_rat,A) )
     => ( ( aa_rat_rat(sgn_sgn_rat,A) != zero_zero_rat )
       => ( ( aa_rat_rat(sgn_sgn_rat,B) != zero_zero_rat )
         => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6538_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6539_sgn__minus__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% sgn_minus_1
tff(fact_6540_sgn__minus__1,axiom,
    aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% sgn_minus_1
tff(fact_6541_sgn__minus__1,axiom,
    aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sgn_minus_1
tff(fact_6542_sgn__minus__1,axiom,
    aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% sgn_minus_1
tff(fact_6543_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_6544_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(M,Xs))),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(N,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_6545_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(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(N,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_6546_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,take_real(M,Xs))),aa_lis3769085826154824284t_real(set_real2,take_real(N,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_6547_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,take_bool(M,Xs))),aa_lis3587183946281582166t_bool(set_bool2,take_bool(N,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_6548_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_set_int_bool(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(N,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_6549_le__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% le_minus_one_simps(1)
tff(fact_6550_le__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% le_minus_one_simps(1)
tff(fact_6551_le__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% le_minus_one_simps(1)
tff(fact_6552_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(3)
tff(fact_6553_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(3)
tff(fact_6554_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(3)
tff(fact_6555_less__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% less_minus_one_simps(1)
tff(fact_6556_less__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% less_minus_one_simps(1)
tff(fact_6557_less__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% less_minus_one_simps(1)
tff(fact_6558_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(3)
tff(fact_6559_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(3)
tff(fact_6560_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(3)
tff(fact_6561_in__image__insert__iff,axiom,
    ! [B3: set_set_VEBT_VEBT,X3: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ! [C6: set_VEBT_VEBT] :
          ( member_set_VEBT_VEBT(C6,B3)
         => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),C6)) )
     => ( member_set_VEBT_VEBT(A2,image_1661326939266726661T_VEBT(insert_VEBT_VEBT(X3),B3))
      <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),A2))
          & member_set_VEBT_VEBT(minus_5127226145743854075T_VEBT(A2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)),B3) ) ) ) ).

% in_image_insert_iff
tff(fact_6562_in__image__insert__iff,axiom,
    ! [B3: set_set_set_nat,X3: set_nat,A2: set_set_nat] :
      ( ! [C6: set_set_nat] :
          ( member_set_set_nat(C6,B3)
         => ~ pp(aa_set_set_nat_bool(member_set_nat(X3),C6)) )
     => ( member_set_set_nat(A2,image_7884819252390400639et_nat(insert_set_nat(X3),B3))
      <=> ( pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
          & member_set_set_nat(minus_2163939370556025621et_nat(A2,aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat)),B3) ) ) ) ).

% in_image_insert_iff
tff(fact_6563_in__image__insert__iff,axiom,
    ! [B3: set_set_real,X3: real,A2: set_real] :
      ( ! [C6: set_real] :
          ( member_set_real(C6,B3)
         => ~ pp(aa_set_real_bool(member_real(X3),C6)) )
     => ( member_set_real(A2,image_2436557299294012491t_real(insert_real(X3),B3))
      <=> ( pp(aa_set_real_bool(member_real(X3),A2))
          & member_set_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)),B3) ) ) ) ).

% in_image_insert_iff
tff(fact_6564_in__image__insert__iff,axiom,
    ! [B3: set_set_bool,X3: bool,A2: set_bool] :
      ( ! [C6: set_bool] :
          ( member_set_bool(C6,B3)
         => ~ pp(aa_set_bool_bool(member_bool(X3),C6)) )
     => ( member_set_bool(A2,image_6568689345319389125t_bool(insert_bool(X3),B3))
      <=> ( pp(aa_set_bool_bool(member_bool(X3),A2))
          & member_set_bool(minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)),B3) ) ) ) ).

% in_image_insert_iff
tff(fact_6565_in__image__insert__iff,axiom,
    ! [B3: set_set_int,X3: int,A2: set_int] :
      ( ! [C6: set_int] :
          ( member_set_int(C6,B3)
         => ~ pp(aa_set_int_bool(member_int(X3),C6)) )
     => ( member_set_int(A2,image_524474410958335435et_int(insert_int(X3),B3))
      <=> ( pp(aa_set_int_bool(member_int(X3),A2))
          & member_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)),B3) ) ) ) ).

% in_image_insert_iff
tff(fact_6566_in__image__insert__iff,axiom,
    ! [B3: set_set_nat,X3: nat,A2: set_nat] :
      ( ! [C6: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(C6),B3))
         => ~ pp(aa_set_nat_bool(member_nat(X3),C6)) )
     => ( pp(aa_set_set_nat_bool(member_set_nat(A2),image_7916887816326733075et_nat(insert_nat(X3),B3)))
      <=> ( pp(aa_set_nat_bool(member_nat(X3),A2))
          & pp(aa_set_set_nat_bool(member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))),B3)) ) ) ) ).

% in_image_insert_iff
tff(fact_6567_abs__minus__le__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A))),zero_z3403309356797280102nteger)) ).

% abs_minus_le_zero
tff(fact_6568_abs__minus__le__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real)) ).

% abs_minus_le_zero
tff(fact_6569_abs__minus__le__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat)) ).

% abs_minus_le_zero
tff(fact_6570_abs__minus__le__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int)) ).

% abs_minus_le_zero
tff(fact_6571_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B) )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6572_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6573_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6574_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6575_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = B )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6576_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6577_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6578_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6579_abs__if__raw,axiom,
    ! [X5: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X5),zero_z3403309356797280102nteger))
       => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X5) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X5) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X5),zero_z3403309356797280102nteger))
       => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_6580_abs__if__raw,axiom,
    ! [X5: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X5) = aa_real_real(uminus_uminus_real,X5) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X5),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_6581_abs__if__raw,axiom,
    ! [X5: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X5) = aa_rat_rat(uminus_uminus_rat,X5) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X5),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_6582_abs__if__raw,axiom,
    ! [X5: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X5) = aa_int_int(uminus_uminus_int,X5) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X5) = X5 ) ) ) ).

% abs_if_raw
tff(fact_6583_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_6584_abs__of__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_6585_abs__of__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_6586_abs__of__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_6587_abs__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A) = A ) ) ) ).

% abs_if
tff(fact_6588_abs__if,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_if
tff(fact_6589_abs__if,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_if
tff(fact_6590_abs__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = A ) ) ) ).

% abs_if
tff(fact_6591_Compl__insert,axiom,
    ! [X3: vEBT_VEBT,A2: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),A2)) = minus_5127226145743854075T_VEBT(uminus8041839845116263051T_VEBT(A2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)) ).

% Compl_insert
tff(fact_6592_Compl__insert,axiom,
    ! [X3: real,A2: set_real] : uminus612125837232591019t_real(aa_set_real_set_real(insert_real(X3),A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) ).

% Compl_insert
tff(fact_6593_Compl__insert,axiom,
    ! [X3: bool,A2: set_bool] : uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(X3),A2)) = minus_minus_set_bool(uminus5166143097222041326t_bool(A2),aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)) ).

% Compl_insert
tff(fact_6594_Compl__insert,axiom,
    ! [X3: int,A2: set_int] : uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(X3),A2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,uminus1532241313380277803et_int(A2)),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) ).

% Compl_insert
tff(fact_6595_Compl__insert,axiom,
    ! [X3: nat,A2: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X3),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) ).

% Compl_insert
tff(fact_6596_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),size_s7982070591426661849_VEBTi(Ys)))
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6597_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),size_s6755466524823107622T_VEBT(Ys)))
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6598_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_real,Ys: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_list_real_nat(size_size_list_real,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_list_real_nat(size_size_list_real,Ys)))
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6599_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_bool,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_list_bool_nat(size_size_list_bool,Ys)))
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
             => ( pp(aa_nat_bool(nth_bool(Xs),I2))
              <=> pp(aa_nat_bool(nth_bool(Ys),I2)) ) )
         => ( take_bool(K,Xs) = take_bool(K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_6600_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_list_nat_nat(size_size_list_nat,Ys)))
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6601_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_int,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_list_int_nat(size_size_list_int,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_list_int_nat(size_size_list_int,Ys)))
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6602_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% sgn_1_neg
tff(fact_6603_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% sgn_1_neg
tff(fact_6604_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% sgn_1_neg
tff(fact_6605_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% sgn_1_neg
tff(fact_6606_sgn__if,axiom,
    ! [X3: code_integer] :
      ( ( ( X3 = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X3) = zero_z3403309356797280102nteger ) )
      & ( ( X3 != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X3))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X3) = one_one_Code_integer ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X3))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X3) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ) ) ).

% sgn_if
tff(fact_6607_sgn__if,axiom,
    ! [X3: real] :
      ( ( ( X3 = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,X3) = zero_zero_real ) )
      & ( ( X3 != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
           => ( aa_real_real(sgn_sgn_real,X3) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
           => ( aa_real_real(sgn_sgn_real,X3) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_if
tff(fact_6608_sgn__if,axiom,
    ! [X3: rat] :
      ( ( ( X3 = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,X3) = zero_zero_rat ) )
      & ( ( X3 != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X3))
           => ( aa_rat_rat(sgn_sgn_rat,X3) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X3))
           => ( aa_rat_rat(sgn_sgn_rat,X3) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_if
tff(fact_6609_sgn__if,axiom,
    ! [X3: int] :
      ( ( ( X3 = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,X3) = zero_zero_int ) )
      & ( ( X3 != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X3))
           => ( aa_int_int(sgn_sgn_int,X3) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X3))
           => ( aa_int_int(sgn_sgn_int,X3) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% sgn_if
tff(fact_6610_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6611_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6612_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6613_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6614_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6615_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6616_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6617_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6618_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6619_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6620_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N) = zero_zero_real )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6621_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6622_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6623_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6624_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6625_zip__replicate2,axiom,
    ! [Xs: list_nat,N: nat,Y: vEBT_VEBT] : zip_nat_VEBT_VEBT(Xs,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),Y)) = map_na3584885621601055599T_VEBT(aTP_Lamm_vl(Y),take_nat(N,Xs)) ).

% zip_replicate2
tff(fact_6626_zip__replicate2,axiom,
    ! [Xs: list_bool,N: nat,Y: vEBT_VEBT] : zip_bool_VEBT_VEBT(Xs,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),Y)) = map_bo7064569317639751025T_VEBT(aTP_Lamm_vj(Y),take_bool(N,Xs)) ).

% zip_replicate2
tff(fact_6627_zip__replicate2,axiom,
    ! [Xs: list_nat,N: nat,Y: bool] : zip_nat_bool(Xs,replicate_bool(N,Y)) = map_na1545282596343781330t_bool(aTP_Lamm_vm(Y),take_nat(N,Xs)) ).

% zip_replicate2
tff(fact_6628_zip__replicate2,axiom,
    ! [Xs: list_bool,N: nat,Y: bool] : zip_bool_bool(Xs,replicate_bool(N,Y)) = map_bo2576003307432440276l_bool(aTP_Lamm_vk(Y),take_bool(N,Xs)) ).

% zip_replicate2
tff(fact_6629_zip__replicate2,axiom,
    ! [Xs: list_nat,N: nat,Y: nat] : zip_nat_nat(Xs,replicate_nat(N,Y)) = map_na7298421622053143531at_nat(aTP_Lamm_vn(Y),take_nat(N,Xs)) ).

% zip_replicate2
tff(fact_6630_zip__replicate2,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,Y: nat] : zip_VEBT_VEBT_nat(Xs,replicate_nat(N,Y)) = map_VE6666121349297532579BT_nat(aTP_Lamm_vf(Y),take_VEBT_VEBT(N,Xs)) ).

% zip_replicate2
tff(fact_6631_zip__replicate2,axiom,
    ! [Xs: list_int,N: nat,Y: int] : zip_int_int(Xs,replicate_int(N,Y)) = map_in7157766398909135175nt_int(aTP_Lamm_wt(Y),take_int(N,Xs)) ).

% zip_replicate2
tff(fact_6632_zip__replicate2,axiom,
    ! [Xs: list_Code_integer,N: nat,Y: code_integer] : zip_Co3543743374963494515nteger(Xs,replic7707675349574490269nteger(N,Y)) = map_Co3589949550033412536nteger(aTP_Lamm_vg(Y),take_Code_integer(N,Xs)) ).

% zip_replicate2
tff(fact_6633_zip__replicate2,axiom,
    ! [Xs: list_nat,N: nat,Y: num] : zip_nat_num(Xs,replicate_num(N,Y)) = map_na8006665559001981237at_num(aTP_Lamm_vo(Y),take_nat(N,Xs)) ).

% zip_replicate2
tff(fact_6634_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_tf,L)),aa_int_set_int(set_or4662586982721622107an_int(zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = aa_int_set_int(set_or4662586982721622107an_int(L),U) ).

% image_add_int_atLeastLessThan
tff(fact_6635_restrict__complement__singleton__eq,axiom,
    ! [F: fun_VE8174548813682830439on_nat,X3: vEBT_VEBT] : restri774867724463461460BT_nat(F,uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) = fun_up5885881570350532375on_nat(F,X3,none_nat) ).

% restrict_complement_singleton_eq
tff(fact_6636_restrict__complement__singleton__eq,axiom,
    ! [F: fun_VE3889768403122177201on_num,X3: vEBT_VEBT] : restri6555571547474015902BT_num(F,uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) = fun_up6594125507299370081on_num(F,X3,none_num) ).

% restrict_complement_singleton_eq
tff(fact_6637_restrict__complement__singleton__eq,axiom,
    ! [F: fun_real_option_nat,X3: real] : restri6827137924477938990al_nat(F,uminus612125837232591019t_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) = fun_up6677080212936659659on_nat(F,X3,none_nat) ).

% restrict_complement_singleton_eq
tff(fact_6638_restrict__complement__singleton__eq,axiom,
    ! [F: fun_real_option_num,X3: real] : restri3384469710633717624al_num(F,uminus612125837232591019t_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) = fun_up7385324149885497365on_num(F,X3,none_num) ).

% restrict_complement_singleton_eq
tff(fact_6639_restrict__complement__singleton__eq,axiom,
    ! [F: fun_bool_option_nat,X3: bool] : restri5405821841207159351ol_nat(F,uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) = fun_up4506005684929255546on_nat(F,X3,none_nat) ).

% restrict_complement_singleton_eq
tff(fact_6640_restrict__complement__singleton__eq,axiom,
    ! [F: fun_bool_option_num,X3: bool] : restri1963153627362937985ol_num(F,uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) = fun_up5214249621878093252on_num(F,X3,none_num) ).

% restrict_complement_singleton_eq
tff(fact_6641_restrict__complement__singleton__eq,axiom,
    ! [F: fun_nat_option_nat,X3: nat] : restrict_map_nat_nat(F,aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) = fun_up1493157387958331631on_nat(F,X3,none_nat) ).

% restrict_complement_singleton_eq
tff(fact_6642_restrict__complement__singleton__eq,axiom,
    ! [F: fun_nat_option_num,X3: nat] : restrict_map_nat_num(F,aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) = fun_up2201401324907169337on_num(F,X3,none_num) ).

% restrict_complement_singleton_eq
tff(fact_6643_restrict__complement__singleton__eq,axiom,
    ! [F: fun_int_option_nat,X3: int] : restrict_map_int_nat(F,uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) = fun_up3620524117960394059on_nat(F,X3,none_nat) ).

% restrict_complement_singleton_eq
tff(fact_6644_restrict__complement__singleton__eq,axiom,
    ! [F: fun_int_option_num,X3: int] : restrict_map_int_num(F,uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) = fun_up4328768054909231765on_num(F,X3,none_num) ).

% restrict_complement_singleton_eq
tff(fact_6645_map__upd__eq__restrict,axiom,
    ! [M: fun_VE8174548813682830439on_nat,X3: vEBT_VEBT] : fun_up5885881570350532375on_nat(M,X3,none_nat) = restri774867724463461460BT_nat(M,uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) ).

% map_upd_eq_restrict
tff(fact_6646_map__upd__eq__restrict,axiom,
    ! [M: fun_VE3889768403122177201on_num,X3: vEBT_VEBT] : fun_up6594125507299370081on_num(M,X3,none_num) = restri6555571547474015902BT_num(M,uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) ).

% map_upd_eq_restrict
tff(fact_6647_map__upd__eq__restrict,axiom,
    ! [M: fun_real_option_nat,X3: real] : fun_up6677080212936659659on_nat(M,X3,none_nat) = restri6827137924477938990al_nat(M,uminus612125837232591019t_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) ).

% map_upd_eq_restrict
tff(fact_6648_map__upd__eq__restrict,axiom,
    ! [M: fun_real_option_num,X3: real] : fun_up7385324149885497365on_num(M,X3,none_num) = restri3384469710633717624al_num(M,uminus612125837232591019t_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) ).

% map_upd_eq_restrict
tff(fact_6649_map__upd__eq__restrict,axiom,
    ! [M: fun_bool_option_nat,X3: bool] : fun_up4506005684929255546on_nat(M,X3,none_nat) = restri5405821841207159351ol_nat(M,uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) ).

% map_upd_eq_restrict
tff(fact_6650_map__upd__eq__restrict,axiom,
    ! [M: fun_bool_option_num,X3: bool] : fun_up5214249621878093252on_num(M,X3,none_num) = restri1963153627362937985ol_num(M,uminus5166143097222041326t_bool(aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))) ).

% map_upd_eq_restrict
tff(fact_6651_map__upd__eq__restrict,axiom,
    ! [M: fun_nat_option_nat,X3: nat] : fun_up1493157387958331631on_nat(M,X3,none_nat) = restrict_map_nat_nat(M,aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) ).

% map_upd_eq_restrict
tff(fact_6652_map__upd__eq__restrict,axiom,
    ! [M: fun_nat_option_num,X3: nat] : fun_up2201401324907169337on_num(M,X3,none_num) = restrict_map_nat_num(M,aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) ).

% map_upd_eq_restrict
tff(fact_6653_map__upd__eq__restrict,axiom,
    ! [M: fun_int_option_nat,X3: int] : fun_up3620524117960394059on_nat(M,X3,none_nat) = restrict_map_int_nat(M,uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) ).

% map_upd_eq_restrict
tff(fact_6654_map__upd__eq__restrict,axiom,
    ! [M: fun_int_option_num,X3: int] : fun_up4328768054909231765on_num(M,X3,none_num) = restrict_map_int_num(M,uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) ).

% map_upd_eq_restrict
tff(fact_6655_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_6656_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_6657_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_6658_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_6659_max__ins__scaled,axiom,
    ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_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,N),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_6660_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(T),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6661_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_6662_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_6663_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_6664_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_6665_VEBT__internal_Oheight_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,X3) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Summary2),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_6666_last__take__nth__conv,axiom,
    ! [N: nat,L: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s7982070591426661849_VEBTi(L)))
     => ( ( N != zero_zero_nat )
       => ( last_VEBT_VEBTi(take_VEBT_VEBTi(N,L)) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_6667_last__take__nth__conv,axiom,
    ! [N: nat,L: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s6755466524823107622T_VEBT(L)))
     => ( ( N != zero_zero_nat )
       => ( last_VEBT_VEBT(take_VEBT_VEBT(N,L)) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_6668_last__take__nth__conv,axiom,
    ! [N: nat,L: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_real_nat(size_size_list_real,L)))
     => ( ( N != zero_zero_nat )
       => ( last_real(take_real(N,L)) = aa_nat_real(nth_real(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_6669_last__take__nth__conv,axiom,
    ! [N: nat,L: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_bool_nat(size_size_list_bool,L)))
     => ( ( N != zero_zero_nat )
       => ( pp(aa_list_bool_bool(last_bool,take_bool(N,L)))
        <=> pp(aa_nat_bool(nth_bool(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ) ).

% last_take_nth_conv
tff(fact_6670_last__take__nth__conv,axiom,
    ! [N: nat,L: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_nat_nat(size_size_list_nat,L)))
     => ( ( N != zero_zero_nat )
       => ( aa_list_nat_nat(last_nat,take_nat(N,L)) = aa_nat_nat(nth_nat(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_6671_last__take__nth__conv,axiom,
    ! [N: nat,L: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_int_nat(size_size_list_int,L)))
     => ( ( N != zero_zero_nat )
       => ( last_int(take_int(N,L)) = aa_nat_int(nth_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_6672_VEBT__internal_Oheight_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),X3))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = zero_zero_nat )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2))) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Summary2),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))))) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_6673_fold__union__pair,axiom,
    ! [B3: set_nat,X3: nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B3))
     => ( sup_su6327502436637775413at_nat(comple5685304695842803022at_nat(image_7178329752028323786at_nat(aTP_Lamm_wu(X3),B3)),A2) = finite3745491028973389255at_nat(aTP_Lamm_wv(X3),A2,B3) ) ) ).

% fold_union_pair
tff(fact_6674_fold__union__pair,axiom,
    ! [B3: set_nat,X3: vEBT_VEBT,A2: set_Pr7556676689462069481BT_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,B3))
     => ( sup_su6061789376821058069BT_nat(comple9061401370350521660BT_nat(image_2578358110157900224BT_nat(aTP_Lamm_ww(X3),B3)),A2) = finite1392145723888408323BT_nat(aTP_Lamm_wx(X3),A2,B3) ) ) ).

% fold_union_pair
tff(fact_6675_fold__union__pair,axiom,
    ! [B3: set_int,X3: int,A2: set_Pr958786334691620121nt_int] :
      ( pp(aa_set_int_bool(finite_finite_int,B3))
     => ( sup_su6024340866399070445nt_int(comple5382143125604098054nt_int(image_8635204845542730022nt_int(aTP_Lamm_wy(X3),B3)),A2) = finite5202366122487795491nt_int(aTP_Lamm_wz(X3),A2,B3) ) ) ).

% fold_union_pair
tff(fact_6676_fold__union__pair,axiom,
    ! [B3: set_Code_integer,X3: code_integer,A2: set_Pr4811707699266497531nteger] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
     => ( sup_su3575575879904067535nteger(comple2203973573673791208nteger(image_8529064383210301591nteger(aTP_Lamm_xa(X3),B3)),A2) = finite6473783728727339668nteger(aTP_Lamm_xb(X3),A2,B3) ) ) ).

% fold_union_pair
tff(fact_6677_fold__union__pair,axiom,
    ! [B3: set_num,X3: nat,A2: set_Pr6200539531224447659at_num] :
      ( pp(aa_set_num_bool(finite_finite_num,B3))
     => ( sup_su2042722026077122175at_num(comple1400524285282149784at_num(image_4778453735051108682at_num(aTP_Lamm_xc(X3),B3)),A2) = finite1345615011996174151at_num(aTP_Lamm_xd(X3),A2,B3) ) ) ).

% fold_union_pair
tff(fact_6678_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_6679_real__add__minus__iff,axiom,
    ! [X3: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( X3 = A ) ) ).

% real_add_minus_iff
tff(fact_6680_negative__zle,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zle
tff(fact_6681_cSup__singleton,axiom,
    ! [X3: bool] :
      ( pp(aa_set_bool_bool(comple6445916836919887748p_bool,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)))
    <=> pp(X3) ) ).

% cSup_singleton
tff(fact_6682_cSup__singleton,axiom,
    ! [X3: int] : aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) = X3 ).

% cSup_singleton
tff(fact_6683_cSup__singleton,axiom,
    ! [X3: nat] : aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) = X3 ).

% cSup_singleton
tff(fact_6684_cSup__singleton,axiom,
    ! [X3: set_nat] : comple7399068483239264473et_nat(aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat)) = X3 ).

% cSup_singleton
tff(fact_6685_cSup__singleton,axiom,
    ! [X3: real] : comple1385675409528146559p_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) = X3 ).

% cSup_singleton
tff(fact_6686_cSup__atLeastLessThan,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3))
     => ( comple1385675409528146559p_real(aa_real_set_real(set_or66887138388493659n_real(Y),X3)) = X3 ) ) ).

% cSup_atLeastLessThan
tff(fact_6687_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zless
tff(fact_6688_Max__singleton,axiom,
    ! [X3: real] : lattic4275903605611617917x_real(aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) = X3 ).

% Max_singleton
tff(fact_6689_Max__singleton,axiom,
    ! [X3: bool] :
      ( pp(aa_set_bool_bool(lattic7308977977949639686x_bool,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)))
    <=> pp(X3) ) ).

% Max_singleton
tff(fact_6690_Max__singleton,axiom,
    ! [X3: nat] : aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) = X3 ).

% Max_singleton
tff(fact_6691_Max__singleton,axiom,
    ! [X3: int] : aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) = X3 ).

% Max_singleton
tff(fact_6692_last__replicate,axiom,
    ! [N: nat,X3: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_list_nat_nat(last_nat,replicate_nat(N,X3)) = X3 ) ) ).

% last_replicate
tff(fact_6693_last__replicate,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( last_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3)) = X3 ) ) ).

% last_replicate
tff(fact_6694_last__replicate,axiom,
    ! [N: nat,X3: bool] :
      ( ( N != zero_zero_nat )
     => ( pp(aa_list_bool_bool(last_bool,replicate_bool(N,X3)))
      <=> pp(X3) ) ) ).

% last_replicate
tff(fact_6695_cSUP__const,axiom,
    ! [A2: set_nat,C: int] :
      ( ( A2 != bot_bot_set_nat )
     => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(aTP_Lamm_xe(C)),A2)) = C ) ) ).

% cSUP_const
tff(fact_6696_cSUP__const,axiom,
    ! [A2: set_int,C: int] :
      ( ( A2 != bot_bot_set_int )
     => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_xf,C)),A2)) = C ) ) ).

% cSUP_const
tff(fact_6697_cSUP__const,axiom,
    ! [A2: set_real,C: nat] :
      ( ( A2 != bot_bot_set_real )
     => ( aa_set_nat_nat(complete_Sup_Sup_nat,image_real_nat(aTP_Lamm_va(C),A2)) = C ) ) ).

% cSUP_const
tff(fact_6698_cSUP__const,axiom,
    ! [A2: set_bool,C: nat] :
      ( ( A2 != bot_bot_set_bool )
     => ( aa_set_nat_nat(complete_Sup_Sup_nat,image_bool_nat(aTP_Lamm_vd(C),A2)) = C ) ) ).

% cSUP_const
tff(fact_6699_cSUP__const,axiom,
    ! [A2: set_nat,C: nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_im,C)),A2)) = C ) ) ).

% cSUP_const
tff(fact_6700_cSUP__const,axiom,
    ! [A2: set_int,C: nat] :
      ( ( A2 != bot_bot_set_int )
     => ( aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat(aTP_Lamm_xg(C),A2)) = C ) ) ).

% cSUP_const
tff(fact_6701_cSUP__const,axiom,
    ! [A2: set_real,C: real] :
      ( ( A2 != bot_bot_set_real )
     => ( comple1385675409528146559p_real(image_real_real(aTP_Lamm_ux(C),A2)) = C ) ) ).

% cSUP_const
tff(fact_6702_cSUP__const,axiom,
    ! [A2: set_bool,C: real] :
      ( ( A2 != bot_bot_set_bool )
     => ( comple1385675409528146559p_real(image_bool_real(aTP_Lamm_vc(C),A2)) = C ) ) ).

% cSUP_const
tff(fact_6703_cSUP__const,axiom,
    ! [A2: set_nat,C: real] :
      ( ( A2 != bot_bot_set_nat )
     => ( comple1385675409528146559p_real(aa_set_nat_set_real(image_nat_real(aTP_Lamm_uw(C)),A2)) = C ) ) ).

% cSUP_const
tff(fact_6704_cSUP__const,axiom,
    ! [A2: set_int,C: real] :
      ( ( A2 != bot_bot_set_int )
     => ( comple1385675409528146559p_real(image_int_real(aTP_Lamm_uy(C),A2)) = C ) ) ).

% cSUP_const
tff(fact_6705_finite__UN__I,axiom,
    ! [A2: set_VEBT_VEBT,B3: fun_VE6538172513902245929et_int] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A2))
           => pp(aa_set_int_bool(finite_finite_int,aa_VEBT_VEBT_set_int(B3,A3))) )
       => pp(aa_set_int_bool(finite_finite_int,comple3221217463730067765et_int(image_2273570491937255121et_int(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6706_finite__UN__I,axiom,
    ! [A2: set_real,B3: fun_real_set_int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => pp(aa_set_int_bool(finite_finite_int,aa_real_set_int(B3,A3))) )
       => pp(aa_set_int_bool(finite_finite_int,comple3221217463730067765et_int(image_real_set_int(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6707_finite__UN__I,axiom,
    ! [A2: set_VEBT_VEBT,B3: fun_VE7607429302209253035omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A2))
           => pp(aa_set_complex_bool(finite3207457112153483333omplex,aa_VEB6977773583474748892omplex(B3,A3))) )
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,comple8424636186594484919omplex(image_7711551513561399379omplex(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6708_finite__UN__I,axiom,
    ! [A2: set_real,B3: fun_real_set_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => pp(aa_set_complex_bool(finite3207457112153483333omplex,aa_real_set_complex(B3,A3))) )
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,comple8424636186594484919omplex(image_2129611632225307415omplex(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6709_finite__UN__I,axiom,
    ! [A2: set_VEBT_VEBT,B3: fun_VE5193490835263103322nteger] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A2))
           => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,aa_VEB8518286878723502091nteger(B3,A3))) )
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,comple739944243200306918nteger(image_7429379093677266050nteger(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6710_finite__UN__I,axiom,
    ! [A2: set_real,B3: fun_re836941188292770620nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,aa_rea3998726609137973757nteger(B3,A3))) )
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,comple739944243200306918nteger(image_1541721294081005382nteger(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6711_finite__UN__I,axiom,
    ! [A2: set_nat,B3: fun_nat_set_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => pp(aa_set_int_bool(finite_finite_int,aa_nat_set_int(B3,A3))) )
       => pp(aa_set_int_bool(finite_finite_int,comple3221217463730067765et_int(image_nat_set_int(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6712_finite__UN__I,axiom,
    ! [A2: set_nat,B3: fun_nat_set_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => pp(aa_set_complex_bool(finite3207457112153483333omplex,aa_nat_set_complex(B3,A3))) )
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,comple8424636186594484919omplex(image_6594795319511438139omplex(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6713_finite__UN__I,axiom,
    ! [A2: set_nat,B3: fun_na9118744110877893344nteger] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,aa_nat8922193080109606305nteger(B3,A3))) )
       => pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,comple739944243200306918nteger(image_2385391725447763818nteger(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6714_finite__UN__I,axiom,
    ! [A2: set_int,B3: fun_int_set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => pp(aa_set_int_bool(finite_finite_int,aa_int_set_int(B3,A3))) )
       => pp(aa_set_int_bool(finite_finite_int,comple3221217463730067765et_int(image_int_set_int(B3,A2)))) ) ) ).

% finite_UN_I
tff(fact_6715_Max_Obounded__iff,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A2)),X3))
        <=> ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X4),X3)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6716_Max_Obounded__iff,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,lattic4275903605611617917x_real(A2)),X3))
        <=> ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),X3)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6717_Max_Obounded__iff,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_set_bool_bool(lattic7308977977949639686x_bool,A2)),X3))
        <=> ! [X4: bool] :
              ( pp(aa_set_bool_bool(member_bool(X4),A2))
             => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X4),X3)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6718_Max_Obounded__iff,axiom,
    ! [A2: set_rat,X3: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,A2)),X3))
        <=> ! [X4: rat] :
              ( pp(aa_set_rat_bool(member_rat(X4),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),X3)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6719_Max_Obounded__iff,axiom,
    ! [A2: set_num,X3: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_set_num_num(lattic4823215512031491691ax_num,A2)),X3))
        <=> ! [X4: num] :
              ( pp(aa_set_num_bool(member_num(X4),A2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),X3)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6720_Max_Obounded__iff,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)),X3))
        <=> ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),X3)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6721_Max_Obounded__iff,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(lattic8263393255366662781ax_int,A2)),X3))
        <=> ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),X3)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6722_Max__less__iff,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,lattic4901227151466704046nteger(A2)),X3))
        <=> ! [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),X3)) ) ) ) ) ).

% Max_less_iff
tff(fact_6723_Max__less__iff,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,aa_set_bool_bool(lattic7308977977949639686x_bool,A2)),X3))
        <=> ! [X4: bool] :
              ( pp(aa_set_bool_bool(member_bool(X4),A2))
             => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X4),X3)) ) ) ) ) ).

% Max_less_iff
tff(fact_6724_Max__less__iff,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,lattic4275903605611617917x_real(A2)),X3))
        <=> ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),X3)) ) ) ) ) ).

% Max_less_iff
tff(fact_6725_Max__less__iff,axiom,
    ! [A2: set_rat,X3: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,A2)),X3))
        <=> ! [X4: rat] :
              ( pp(aa_set_rat_bool(member_rat(X4),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),X3)) ) ) ) ) ).

% Max_less_iff
tff(fact_6726_Max__less__iff,axiom,
    ! [A2: set_num,X3: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_set_num_num(lattic4823215512031491691ax_num,A2)),X3))
        <=> ! [X4: num] :
              ( pp(aa_set_num_bool(member_num(X4),A2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),X3)) ) ) ) ) ).

% Max_less_iff
tff(fact_6727_Max__less__iff,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)),X3))
        <=> ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),X3)) ) ) ) ) ).

% Max_less_iff
tff(fact_6728_Max__less__iff,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_set_int_int(lattic8263393255366662781ax_int,A2)),X3))
        <=> ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),X3)) ) ) ) ) ).

% Max_less_iff
tff(fact_6729_Max__const,axiom,
    ! [A2: set_nat,C: real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( lattic4275903605611617917x_real(aa_set_nat_set_real(image_nat_real(aTP_Lamm_uw(C)),A2)) = C ) ) ) ).

% Max_const
tff(fact_6730_Max__const,axiom,
    ! [A2: set_complex,C: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,image_complex_nat(aTP_Lamm_xh(C),A2)) = C ) ) ) ).

% Max_const
tff(fact_6731_Max__const,axiom,
    ! [A2: set_Code_integer,C: nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,image_951025933927791156er_nat(aTP_Lamm_xi(C),A2)) = C ) ) ) ).

% Max_const
tff(fact_6732_Max__const,axiom,
    ! [A2: set_real,C: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,image_real_nat(aTP_Lamm_va(C),A2)) = C ) ) ) ).

% Max_const
tff(fact_6733_Max__const,axiom,
    ! [A2: set_bool,C: nat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,image_bool_nat(aTP_Lamm_vd(C),A2)) = C ) ) ) ).

% Max_const
tff(fact_6734_Max__const,axiom,
    ! [A2: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_im,C)),A2)) = C ) ) ) ).

% Max_const
tff(fact_6735_Max__const,axiom,
    ! [A2: set_int,C: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,image_int_nat(aTP_Lamm_xg(C),A2)) = C ) ) ) ).

% Max_const
tff(fact_6736_Max__const,axiom,
    ! [A2: set_complex,C: int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( aa_set_int_int(lattic8263393255366662781ax_int,image_complex_int(aTP_Lamm_xj(C),A2)) = C ) ) ) ).

% Max_const
tff(fact_6737_Max__const,axiom,
    ! [A2: set_Code_integer,C: int] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( aa_set_int_int(lattic8263393255366662781ax_int,image_948535463418740880er_int(aTP_Lamm_xk(C),A2)) = C ) ) ) ).

% Max_const
tff(fact_6738_Max__const,axiom,
    ! [A2: set_real,C: int] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( aa_set_int_int(lattic8263393255366662781ax_int,image_real_int(aTP_Lamm_vb(C),A2)) = C ) ) ) ).

% Max_const
tff(fact_6739_set__concat,axiom,
    ! [Xs: list_list_VEBT_VEBT] : aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,concat_VEBT_VEBT(Xs)) = comple2820511241208326657T_VEBT(image_6463372868993444447T_VEBT(set_VEBT_VEBT2,set_list_VEBT_VEBT2(Xs))) ).

% set_concat
tff(fact_6740_set__concat,axiom,
    ! [Xs: list_list_real] : aa_lis3769085826154824284t_real(set_real2,concat_real(Xs)) = comple3096694443085538997t_real(image_6239767680843085477t_real(set_real2,set_list_real2(Xs))) ).

% set_concat
tff(fact_6741_set__concat,axiom,
    ! [Xs: list_list_bool] : aa_lis3587183946281582166t_bool(set_bool2,concat_bool(Xs)) = comple9167723193526752868t_bool(image_9119081611755079455t_bool(set_bool2,set_list_bool2(Xs))) ).

% set_concat
tff(fact_6742_set__concat,axiom,
    ! [Xs: list_list_nat] : aa_list_nat_set_nat(set_nat2,concat_nat(Xs)) = comple7399068483239264473et_nat(image_1775855109352712557et_nat(set_nat2,set_list_nat2(Xs))) ).

% set_concat
tff(fact_6743_Some__Sup,axiom,
    ! [A2: set_bool] :
      ( ( A2 != bot_bot_set_bool )
     => ( aa_bool_option_bool(some_bool,aa_set_bool_bool(comple6445916836919887748p_bool,A2)) = comple4504012738199311306n_bool(image_6141813941430646411n_bool(some_bool,A2)) ) ) ).

% Some_Sup
tff(fact_6744_Some__Sup,axiom,
    ! [A2: set_set_nat] :
      ( ( A2 != bot_bot_set_set_nat )
     => ( aa_set6119337032432945050et_nat(some_set_nat,comple7399068483239264473et_nat(A2)) = comple8455683388168444585et_nat(image_838307146230912995et_nat(some_set_nat,A2)) ) ) ).

% Some_Sup
tff(fact_6745_cSup__eq__Max,axiom,
    ! [X6: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,X6))
     => ( ( X6 != bot_bot_set_real )
       => ( comple1385675409528146559p_real(X6) = lattic4275903605611617917x_real(X6) ) ) ) ).

% cSup_eq_Max
tff(fact_6746_cSup__eq__Max,axiom,
    ! [X6: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,X6))
     => ( ( X6 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X6) ) ) ) ).

% cSup_eq_Max
tff(fact_6747_cSup__eq__Max,axiom,
    ! [X6: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,X6))
     => ( ( X6 != bot_bot_set_int )
       => ( aa_set_int_int(complete_Sup_Sup_int,X6) = aa_set_int_int(lattic8263393255366662781ax_int,X6) ) ) ) ).

% cSup_eq_Max
tff(fact_6748_cSup__eq__maximum,axiom,
    ! [Z: set_int,X6: set_set_int] :
      ( member_set_int(Z,X6)
     => ( ! [X: set_int] :
            ( member_set_int(X,X6)
           => pp(aa_set_int_bool(ord_less_eq_set_int(X),Z)) )
       => ( comple3221217463730067765et_int(X6) = Z ) ) ) ).

% cSup_eq_maximum
tff(fact_6749_cSup__eq__maximum,axiom,
    ! [Z: int,X6: set_int] :
      ( pp(aa_set_int_bool(member_int(Z),X6))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),X6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) )
       => ( aa_set_int_int(complete_Sup_Sup_int,X6) = Z ) ) ) ).

% cSup_eq_maximum
tff(fact_6750_cSup__eq__maximum,axiom,
    ! [Z: nat,X6: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(Z),X6))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),X6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = Z ) ) ) ).

% cSup_eq_maximum
tff(fact_6751_cSup__eq__maximum,axiom,
    ! [Z: set_nat,X6: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(Z),X6))
     => ( ! [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),X6))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Z)) )
       => ( comple7399068483239264473et_nat(X6) = Z ) ) ) ).

% cSup_eq_maximum
tff(fact_6752_cSup__eq__maximum,axiom,
    ! [Z: real,X6: set_real] :
      ( pp(aa_set_real_bool(member_real(Z),X6))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),X6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Z)) )
       => ( comple1385675409528146559p_real(X6) = Z ) ) ) ).

% cSup_eq_maximum
tff(fact_6753_cSup__eq,axiom,
    ! [X6: set_int,A: int] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),X6))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A)) )
     => ( ! [Y3: int] :
            ( ! [X5: int] :
                ( pp(aa_set_int_bool(member_int(X5),X6))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Y3)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),Y3)) )
       => ( aa_set_int_int(complete_Sup_Sup_int,X6) = A ) ) ) ).

% cSup_eq
tff(fact_6754_cSup__eq,axiom,
    ! [X6: set_real,A: real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),X6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A)) )
     => ( ! [Y3: real] :
            ( ! [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),X6))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Y3)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y3)) )
       => ( comple1385675409528146559p_real(X6) = A ) ) ) ).

% cSup_eq
tff(fact_6755_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_6756_uminus__set__def,axiom,
    ! [A2: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(A2) = collect_VEBT_VEBT(uminus8258794197550912879T_bool(aa_set1728193350978918526T_bool(aTP_Lamm_an,A2))) ).

% uminus_set_def
tff(fact_6757_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus4638318004487240379l_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ao,A2))) ).

% uminus_set_def
tff(fact_6758_uminus__set__def,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(uminus8337675090112669241x_bool(aTP_Lamm_ap(A2))) ).

% uminus_set_def
tff(fact_6759_uminus__set__def,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(uminus5119773305880778631t_bool(aTP_Lamm_aq(A2))) ).

% uminus_set_def
tff(fact_6760_uminus__set__def,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(uminus6226608070181861729t_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ar,A2))) ).

% uminus_set_def
tff(fact_6761_uminus__set__def,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2) = collect_nat(uminus1995910650241132439t_bool(aa_set3744802623701456604t_bool(aTP_Lamm_as,A2))) ).

% uminus_set_def
tff(fact_6762_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus8094991720627284027t_bool(aa_set7012339369581096484t_bool(aTP_Lamm_at,A2))) ).

% uminus_set_def
tff(fact_6763_Some__SUP,axiom,
    ! [A2: set_real,F: fun_real_set_nat] :
      ( ( A2 != bot_bot_set_real )
     => ( aa_set6119337032432945050et_nat(some_set_nat,comple7399068483239264473et_nat(image_real_set_nat(F,A2))) = comple8455683388168444585et_nat(image_2508973409438724873et_nat(aTP_Lamm_xl(F),A2)) ) ) ).

% Some_SUP
tff(fact_6764_Some__SUP,axiom,
    ! [A2: set_bool,F: fun_bool_set_nat] :
      ( ( A2 != bot_bot_set_bool )
     => ( aa_set6119337032432945050et_nat(some_set_nat,comple7399068483239264473et_nat(image_bool_set_nat(F,A2))) = comple8455683388168444585et_nat(image_4039916617228794792et_nat(aTP_Lamm_xm(F),A2)) ) ) ).

% Some_SUP
tff(fact_6765_Some__SUP,axiom,
    ! [A2: set_nat,F: fun_nat_set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( aa_set6119337032432945050et_nat(some_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))) = comple8455683388168444585et_nat(image_1085372205585808685et_nat(aTP_Lamm_xn(F),A2)) ) ) ).

% Some_SUP
tff(fact_6766_Some__SUP,axiom,
    ! [A2: set_int,F: fun_int_set_nat] :
      ( ( A2 != bot_bot_set_int )
     => ( aa_set6119337032432945050et_nat(some_set_nat,comple7399068483239264473et_nat(image_int_set_nat(F,A2))) = comple8455683388168444585et_nat(image_2293489338514188681et_nat(aTP_Lamm_xo(F),A2)) ) ) ).

% Some_SUP
tff(fact_6767_cSup__eq__non__empty,axiom,
    ! [X6: set_bool,A: bool] :
      ( ( X6 != bot_bot_set_bool )
     => ( ! [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),X6))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X),A)) )
       => ( ! [Y3: bool] :
              ( ! [X5: bool] :
                  ( pp(aa_set_bool_bool(member_bool(X5),X6))
                 => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X5),Y3)) )
             => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,A),Y3)) )
         => ( pp(aa_set_bool_bool(comple6445916836919887748p_bool,X6))
          <=> pp(A) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_6768_cSup__eq__non__empty,axiom,
    ! [X6: set_set_int,A: set_int] :
      ( ( X6 != bot_bot_set_set_int )
     => ( ! [X: set_int] :
            ( member_set_int(X,X6)
           => pp(aa_set_int_bool(ord_less_eq_set_int(X),A)) )
       => ( ! [Y3: set_int] :
              ( ! [X5: set_int] :
                  ( member_set_int(X5,X6)
                 => pp(aa_set_int_bool(ord_less_eq_set_int(X5),Y3)) )
             => pp(aa_set_int_bool(ord_less_eq_set_int(A),Y3)) )
         => ( comple3221217463730067765et_int(X6) = A ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_6769_cSup__eq__non__empty,axiom,
    ! [X6: set_int,A: int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),X6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A)) )
       => ( ! [Y3: int] :
              ( ! [X5: int] :
                  ( pp(aa_set_int_bool(member_int(X5),X6))
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Y3)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),Y3)) )
         => ( aa_set_int_int(complete_Sup_Sup_int,X6) = A ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_6770_cSup__eq__non__empty,axiom,
    ! [X6: set_nat,A: nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),X6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A)) )
       => ( ! [Y3: nat] :
              ( ! [X5: nat] :
                  ( pp(aa_set_nat_bool(member_nat(X5),X6))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y3)) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),Y3)) )
         => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = A ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_6771_cSup__eq__non__empty,axiom,
    ! [X6: set_set_nat,A: set_nat] :
      ( ( X6 != bot_bot_set_set_nat )
     => ( ! [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),X6))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),A)) )
       => ( ! [Y3: set_nat] :
              ( ! [X5: set_nat] :
                  ( pp(aa_set_set_nat_bool(member_set_nat(X5),X6))
                 => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X5),Y3)) )
             => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A),Y3)) )
         => ( comple7399068483239264473et_nat(X6) = A ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_6772_cSup__eq__non__empty,axiom,
    ! [X6: set_real,A: real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),X6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A)) )
       => ( ! [Y3: real] :
              ( ! [X5: real] :
                  ( pp(aa_set_real_bool(member_real(X5),X6))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X5),Y3)) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y3)) )
         => ( comple1385675409528146559p_real(X6) = A ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_6773_cSup__least,axiom,
    ! [X6: set_bool,Z: bool] :
      ( ( X6 != bot_bot_set_bool )
     => ( ! [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),X6))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X),Z)) )
       => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_set_bool_bool(comple6445916836919887748p_bool,X6)),Z)) ) ) ).

% cSup_least
tff(fact_6774_cSup__least,axiom,
    ! [X6: set_set_int,Z: set_int] :
      ( ( X6 != bot_bot_set_set_int )
     => ( ! [X: set_int] :
            ( member_set_int(X,X6)
           => pp(aa_set_int_bool(ord_less_eq_set_int(X),Z)) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(X6)),Z)) ) ) ).

% cSup_least
tff(fact_6775_cSup__least,axiom,
    ! [X6: set_int,Z: int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),X6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,X6)),Z)) ) ) ).

% cSup_least
tff(fact_6776_cSup__least,axiom,
    ! [X6: set_nat,Z: nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),X6))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,X6)),Z)) ) ) ).

% cSup_least
tff(fact_6777_cSup__least,axiom,
    ! [X6: set_set_nat,Z: set_nat] :
      ( ( X6 != bot_bot_set_set_nat )
     => ( ! [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),X6))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Z)) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(X6)),Z)) ) ) ).

% cSup_least
tff(fact_6778_cSup__least,axiom,
    ! [X6: set_real,Z: real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),X6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Z)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,comple1385675409528146559p_real(X6)),Z)) ) ) ).

% cSup_least
tff(fact_6779_le__cSup__finite,axiom,
    ! [X6: set_set_int,X3: set_int] :
      ( finite6197958912794628473et_int(X6)
     => ( member_set_int(X3,X6)
       => pp(aa_set_int_bool(ord_less_eq_set_int(X3),comple3221217463730067765et_int(X6))) ) ) ).

% le_cSup_finite
tff(fact_6780_le__cSup__finite,axiom,
    ! [X6: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,X6))
     => ( pp(aa_set_int_bool(member_int(X3),X6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_set_int_int(complete_Sup_Sup_int,X6))) ) ) ).

% le_cSup_finite
tff(fact_6781_le__cSup__finite,axiom,
    ! [X6: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,X6))
     => ( pp(aa_set_nat_bool(member_nat(X3),X6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_set_nat_nat(complete_Sup_Sup_nat,X6))) ) ) ).

% le_cSup_finite
tff(fact_6782_le__cSup__finite,axiom,
    ! [X6: set_set_nat,X3: set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,X6))
     => ( pp(aa_set_set_nat_bool(member_set_nat(X3),X6))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),comple7399068483239264473et_nat(X6))) ) ) ).

% le_cSup_finite
tff(fact_6783_le__cSup__finite,axiom,
    ! [X6: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,X6))
     => ( pp(aa_set_real_bool(member_real(X3),X6))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),comple1385675409528146559p_real(X6))) ) ) ).

% le_cSup_finite
tff(fact_6784_less__cSupE,axiom,
    ! [Y: int,X6: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_set_int_int(complete_Sup_Sup_int,X6)))
     => ( ( X6 != bot_bot_set_int )
       => ~ ! [X: int] :
              ( pp(aa_set_int_bool(member_int(X),X6))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ) ).

% less_cSupE
tff(fact_6785_less__cSupE,axiom,
    ! [Y: nat,X6: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_set_nat_nat(complete_Sup_Sup_nat,X6)))
     => ( ( X6 != bot_bot_set_nat )
       => ~ ! [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),X6))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ) ).

% less_cSupE
tff(fact_6786_less__cSupE,axiom,
    ! [Y: real,X6: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),comple1385675409528146559p_real(X6)))
     => ( ( X6 != bot_bot_set_real )
       => ~ ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),X6))
             => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% less_cSupE
tff(fact_6787_less__cSupD,axiom,
    ! [X6: set_int,Z: int] :
      ( ( X6 != bot_bot_set_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_set_int_int(complete_Sup_Sup_int,X6)))
       => ? [X: int] :
            ( pp(aa_set_int_bool(member_int(X),X6))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X)) ) ) ) ).

% less_cSupD
tff(fact_6788_less__cSupD,axiom,
    ! [X6: set_nat,Z: nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),aa_set_nat_nat(complete_Sup_Sup_nat,X6)))
       => ? [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),X6))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X)) ) ) ) ).

% less_cSupD
tff(fact_6789_less__cSupD,axiom,
    ! [X6: set_real,Z: real] :
      ( ( X6 != bot_bot_set_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),comple1385675409528146559p_real(X6)))
       => ? [X: real] :
            ( pp(aa_set_real_bool(member_real(X),X6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X)) ) ) ) ).

% less_cSupD
tff(fact_6790_finite__imp__Sup__less,axiom,
    ! [X6: set_int,X3: int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,X6))
     => ( pp(aa_set_int_bool(member_int(X3),X6))
       => ( ! [X: int] :
              ( pp(aa_set_int_bool(member_int(X),X6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_set_int_int(complete_Sup_Sup_int,X6)),A)) ) ) ) ).

% finite_imp_Sup_less
tff(fact_6791_finite__imp__Sup__less,axiom,
    ! [X6: set_nat,X3: nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,X6))
     => ( pp(aa_set_nat_bool(member_nat(X3),X6))
       => ( ! [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),X6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),A)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(complete_Sup_Sup_nat,X6)),A)) ) ) ) ).

% finite_imp_Sup_less
tff(fact_6792_finite__imp__Sup__less,axiom,
    ! [X6: set_real,X3: real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,X6))
     => ( pp(aa_set_real_bool(member_real(X3),X6))
       => ( ! [X: real] :
              ( pp(aa_set_real_bool(member_real(X),X6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,comple1385675409528146559p_real(X6)),A)) ) ) ) ).

% finite_imp_Sup_less
tff(fact_6793_Max_OcoboundedI,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(A),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),lattic4275903605611617917x_real(A2))) ) ) ).

% Max.coboundedI
tff(fact_6794_Max_OcoboundedI,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),lattic4901227151466704046nteger(A2))) ) ) ).

% Max.coboundedI
tff(fact_6795_Max_OcoboundedI,axiom,
    ! [A2: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(member_rat(A),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_set_rat_rat(lattic7630753665789217321ax_rat,A2))) ) ) ).

% Max.coboundedI
tff(fact_6796_Max_OcoboundedI,axiom,
    ! [A2: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(member_num(A),A2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_set_num_num(lattic4823215512031491691ax_num,A2))) ) ) ).

% Max.coboundedI
tff(fact_6797_Max_OcoboundedI,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_set_nat_nat(lattic8265883725875713057ax_nat,A2))) ) ) ).

% Max.coboundedI
tff(fact_6798_Max_OcoboundedI,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_set_int_int(lattic8263393255366662781ax_int,A2))) ) ) ).

% Max.coboundedI
tff(fact_6799_Max__eq__if,axiom,
    ! [A2: set_Code_integer,B3: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
       => ( ! [X: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),A2))
             => ? [Xa2: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),B3))
                  & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Xa2)) ) )
         => ( ! [X: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X),B3))
               => ? [Xa2: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa2),A2))
                    & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Xa2)) ) )
           => ( lattic4901227151466704046nteger(A2) = lattic4901227151466704046nteger(B3) ) ) ) ) ) ).

% Max_eq_if
tff(fact_6800_Max__eq__if,axiom,
    ! [A2: set_rat,B3: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(finite_finite_rat,B3))
       => ( ! [X: rat] :
              ( pp(aa_set_rat_bool(member_rat(X),A2))
             => ? [Xa2: rat] :
                  ( pp(aa_set_rat_bool(member_rat(Xa2),B3))
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Xa2)) ) )
         => ( ! [X: rat] :
                ( pp(aa_set_rat_bool(member_rat(X),B3))
               => ? [Xa2: rat] :
                    ( pp(aa_set_rat_bool(member_rat(Xa2),A2))
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Xa2)) ) )
           => ( aa_set_rat_rat(lattic7630753665789217321ax_rat,A2) = aa_set_rat_rat(lattic7630753665789217321ax_rat,B3) ) ) ) ) ) ).

% Max_eq_if
tff(fact_6801_Max__eq__if,axiom,
    ! [A2: set_num,B3: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(finite_finite_num,B3))
       => ( ! [X: num] :
              ( pp(aa_set_num_bool(member_num(X),A2))
             => ? [Xa2: num] :
                  ( pp(aa_set_num_bool(member_num(Xa2),B3))
                  & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Xa2)) ) )
         => ( ! [X: num] :
                ( pp(aa_set_num_bool(member_num(X),B3))
               => ? [Xa2: num] :
                    ( pp(aa_set_num_bool(member_num(Xa2),A2))
                    & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Xa2)) ) )
           => ( aa_set_num_num(lattic4823215512031491691ax_num,A2) = aa_set_num_num(lattic4823215512031491691ax_num,B3) ) ) ) ) ) ).

% Max_eq_if
tff(fact_6802_Max__eq__if,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( ! [X: nat] :
              ( pp(aa_set_nat_bool(member_nat(X),A2))
             => ? [Xa2: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa2),B3))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Xa2)) ) )
         => ( ! [X: nat] :
                ( pp(aa_set_nat_bool(member_nat(X),B3))
               => ? [Xa2: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa2),A2))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Xa2)) ) )
           => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A2) = aa_set_nat_nat(lattic8265883725875713057ax_nat,B3) ) ) ) ) ) ).

% Max_eq_if
tff(fact_6803_Max__eq__if,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(finite_finite_int,B3))
       => ( ! [X: int] :
              ( pp(aa_set_int_bool(member_int(X),A2))
             => ? [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),B3))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2)) ) )
         => ( ! [X: int] :
                ( pp(aa_set_int_bool(member_int(X),B3))
               => ? [Xa2: int] :
                    ( pp(aa_set_int_bool(member_int(Xa2),A2))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2)) ) )
           => ( aa_set_int_int(lattic8263393255366662781ax_int,A2) = aa_set_int_int(lattic8263393255366662781ax_int,B3) ) ) ) ) ) ).

% Max_eq_if
tff(fact_6804_Max__eqI,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [Y3: real] :
            ( pp(aa_set_real_bool(member_real(Y3),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),X3)) )
       => ( pp(aa_set_real_bool(member_real(X3),A2))
         => ( lattic4275903605611617917x_real(A2) = X3 ) ) ) ) ).

% Max_eqI
tff(fact_6805_Max__eqI,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ! [Y3: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Y3),A2))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y3),X3)) )
       => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A2))
         => ( lattic4901227151466704046nteger(A2) = X3 ) ) ) ) ).

% Max_eqI
tff(fact_6806_Max__eqI,axiom,
    ! [A2: set_rat,X3: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ! [Y3: rat] :
            ( pp(aa_set_rat_bool(member_rat(Y3),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y3),X3)) )
       => ( pp(aa_set_rat_bool(member_rat(X3),A2))
         => ( aa_set_rat_rat(lattic7630753665789217321ax_rat,A2) = X3 ) ) ) ) ).

% Max_eqI
tff(fact_6807_Max__eqI,axiom,
    ! [A2: set_num,X3: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ! [Y3: num] :
            ( pp(aa_set_num_bool(member_num(Y3),A2))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y3),X3)) )
       => ( pp(aa_set_num_bool(member_num(X3),A2))
         => ( aa_set_num_num(lattic4823215512031491691ax_num,A2) = X3 ) ) ) ) ).

% Max_eqI
tff(fact_6808_Max__eqI,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X3)) )
       => ( pp(aa_set_nat_bool(member_nat(X3),A2))
         => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A2) = X3 ) ) ) ) ).

% Max_eqI
tff(fact_6809_Max__eqI,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [Y3: int] :
            ( pp(aa_set_int_bool(member_int(Y3),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y3),X3)) )
       => ( pp(aa_set_int_bool(member_int(X3),A2))
         => ( aa_set_int_int(lattic8263393255366662781ax_int,A2) = X3 ) ) ) ) ).

% Max_eqI
tff(fact_6810_Max__ge,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X3),A2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),lattic4275903605611617917x_real(A2))) ) ) ).

% Max_ge
tff(fact_6811_Max__ge,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X3),lattic4901227151466704046nteger(A2))) ) ) ).

% Max_ge
tff(fact_6812_Max__ge,axiom,
    ! [A2: set_rat,X3: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(member_rat(X3),A2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_set_rat_rat(lattic7630753665789217321ax_rat,A2))) ) ) ).

% Max_ge
tff(fact_6813_Max__ge,axiom,
    ! [A2: set_num,X3: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(member_num(X3),A2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),aa_set_num_num(lattic4823215512031491691ax_num,A2))) ) ) ).

% Max_ge
tff(fact_6814_Max__ge,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(X3),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_set_nat_nat(lattic8265883725875713057ax_nat,A2))) ) ) ).

% Max_ge
tff(fact_6815_Max__ge,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X3),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_set_int_int(lattic8263393255366662781ax_int,A2))) ) ) ).

% Max_ge
tff(fact_6816_Max__in,axiom,
    ! [A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => pp(aa_set1374477347934578310r_bool(member_Code_integer(lattic4901227151466704046nteger(A2)),A2)) ) ) ).

% Max_in
tff(fact_6817_Max__in,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => pp(aa_set_real_bool(member_real(lattic4275903605611617917x_real(A2)),A2)) ) ) ).

% Max_in
tff(fact_6818_Max__in,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => pp(aa_set_bool_bool(member_bool(aa_set_bool_bool(lattic7308977977949639686x_bool,A2)),A2)) ) ) ).

% Max_in
tff(fact_6819_Max__in,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)),A2)) ) ) ).

% Max_in
tff(fact_6820_Max__in,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => pp(aa_set_int_bool(member_int(aa_set_int_int(lattic8263393255366662781ax_int,A2)),A2)) ) ) ).

% Max_in
tff(fact_6821_int__of__nat__induct,axiom,
    ! [P: fun_int_bool,Z: int] :
      ( ! [N2: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)))))
       => pp(aa_int_bool(P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_6822_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% int_cases
tff(fact_6823_real__minus__mult__self__le,axiom,
    ! [U: real,X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3))) ).

% real_minus_mult_self_le
tff(fact_6824_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_6825_cSUP__least,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,M8: int] :
      ( ( A2 != bot_bo8194388402131092736T_VEBT )
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_VEBT_VEBT_int(F,X)),M8)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_VEBT_VEBT_int(F,A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6826_cSUP__least,axiom,
    ! [A2: set_real,F: fun_real_int,M8: int] :
      ( ( A2 != bot_bot_set_real )
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,X)),M8)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_real_int(F,A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6827_cSUP__least,axiom,
    ! [A2: set_bool,F: fun_bool_int,M8: int] :
      ( ( A2 != bot_bot_set_bool )
     => ( ! [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(F,X)),M8)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_bool_int(F,A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6828_cSUP__least,axiom,
    ! [A2: set_nat,F: fun_nat_int,M8: int] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X)),M8)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6829_cSUP__least,axiom,
    ! [A2: set_int,F: fun_int_int,M8: int] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X)),M8)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6830_cSUP__least,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,M8: nat] :
      ( ( A2 != bot_bo8194388402131092736T_VEBT )
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X)),M8)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_VEBT_VEBT_nat(F,A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6831_cSUP__least,axiom,
    ! [A2: set_real,F: fun_real_nat,M8: nat] :
      ( ( A2 != bot_bot_set_real )
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X)),M8)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_real_nat(F,A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6832_cSUP__least,axiom,
    ! [A2: set_bool,F: fun_bool_nat,M8: nat] :
      ( ( A2 != bot_bot_set_bool )
     => ( ! [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(F,X)),M8)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_bool_nat(F,A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6833_cSUP__least,axiom,
    ! [A2: set_nat,F: fun_nat_nat,M8: nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X)),M8)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6834_cSUP__least,axiom,
    ! [A2: set_int,F: fun_int_nat,M8: nat] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X)),M8)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat(F,A2))),M8)) ) ) ).

% cSUP_least
tff(fact_6835_finite__Sup__less__iff,axiom,
    ! [X6: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,X6))
     => ( ( X6 != bot_bot_set_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_set_int_int(complete_Sup_Sup_int,X6)),A))
        <=> ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),X6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),A)) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_6836_finite__Sup__less__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,X6))
     => ( ( X6 != bot_bot_set_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(complete_Sup_Sup_nat,X6)),A))
        <=> ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),X6))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),A)) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_6837_finite__Sup__less__iff,axiom,
    ! [X6: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,X6))
     => ( ( X6 != bot_bot_set_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,comple1385675409528146559p_real(X6)),A))
        <=> ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),X6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),A)) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_6838_cSup__abs__le,axiom,
    ! [S3: set_int,A: int] :
      ( ( S3 != bot_bot_set_int )
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),S3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),A)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_set_int_int(complete_Sup_Sup_int,S3))),A)) ) ) ).

% cSup_abs_le
tff(fact_6839_cSup__abs__le,axiom,
    ! [S3: set_real,A: real] :
      ( ( S3 != bot_bot_set_real )
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),S3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),A)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,comple1385675409528146559p_real(S3))),A)) ) ) ).

% cSup_abs_le
tff(fact_6840_finite__Sup__in,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( ! [X: bool,Y3: bool] :
              ( pp(aa_set_bool_bool(member_bool(X),A2))
             => ( pp(aa_set_bool_bool(member_bool(Y3),A2))
               => pp(aa_set_bool_bool(member_bool(aa_bool_bool(aa_boo8970175072276269406l_bool(sup_sup_bool,X),Y3)),A2)) ) )
         => pp(aa_set_bool_bool(member_bool(aa_set_bool_bool(comple6445916836919887748p_bool,A2)),A2)) ) ) ) ).

% finite_Sup_in
tff(fact_6841_finite__Sup__in,axiom,
    ! [A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( ( A2 != bot_bot_set_set_nat )
       => ( ! [X: set_nat,Y3: set_nat] :
              ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
             => ( pp(aa_set_set_nat_bool(member_set_nat(Y3),A2))
               => pp(aa_set_set_nat_bool(member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y3)),A2)) ) )
         => pp(aa_set_set_nat_bool(member_set_nat(comple7399068483239264473et_nat(A2)),A2)) ) ) ) ).

% finite_Sup_in
tff(fact_6842_Max_OboundedI,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ! [A3: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A3),A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A3),X3)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A2)),X3)) ) ) ) ).

% Max.boundedI
tff(fact_6843_Max_OboundedI,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [A3: real] :
              ( pp(aa_set_real_bool(member_real(A3),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),X3)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,lattic4275903605611617917x_real(A2)),X3)) ) ) ) ).

% Max.boundedI
tff(fact_6844_Max_OboundedI,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( ! [A3: bool] :
              ( pp(aa_set_bool_bool(member_bool(A3),A2))
             => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,A3),X3)) )
         => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_set_bool_bool(lattic7308977977949639686x_bool,A2)),X3)) ) ) ) ).

% Max.boundedI
tff(fact_6845_Max_OboundedI,axiom,
    ! [A2: set_rat,X3: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( pp(aa_set_rat_bool(member_rat(A3),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A3),X3)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,A2)),X3)) ) ) ) ).

% Max.boundedI
tff(fact_6846_Max_OboundedI,axiom,
    ! [A2: set_num,X3: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ( ! [A3: num] :
              ( pp(aa_set_num_bool(member_num(A3),A2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A3),X3)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_set_num_num(lattic4823215512031491691ax_num,A2)),X3)) ) ) ) ).

% Max.boundedI
tff(fact_6847_Max_OboundedI,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( pp(aa_set_nat_bool(member_nat(A3),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A3),X3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)),X3)) ) ) ) ).

% Max.boundedI
tff(fact_6848_Max_OboundedI,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),X3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(lattic8263393255366662781ax_int,A2)),X3)) ) ) ) ).

% Max.boundedI
tff(fact_6849_Max_OboundedE,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A2)),X3))
         => ! [A10: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A10),A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A10),X3)) ) ) ) ) ).

% Max.boundedE
tff(fact_6850_Max_OboundedE,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,lattic4275903605611617917x_real(A2)),X3))
         => ! [A10: real] :
              ( pp(aa_set_real_bool(member_real(A10),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A10),X3)) ) ) ) ) ).

% Max.boundedE
tff(fact_6851_Max_OboundedE,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_set_bool_bool(lattic7308977977949639686x_bool,A2)),X3))
         => ! [A10: bool] :
              ( pp(aa_set_bool_bool(member_bool(A10),A2))
             => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,A10),X3)) ) ) ) ) ).

% Max.boundedE
tff(fact_6852_Max_OboundedE,axiom,
    ! [A2: set_rat,X3: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,A2)),X3))
         => ! [A10: rat] :
              ( pp(aa_set_rat_bool(member_rat(A10),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A10),X3)) ) ) ) ) ).

% Max.boundedE
tff(fact_6853_Max_OboundedE,axiom,
    ! [A2: set_num,X3: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_set_num_num(lattic4823215512031491691ax_num,A2)),X3))
         => ! [A10: num] :
              ( pp(aa_set_num_bool(member_num(A10),A2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A10),X3)) ) ) ) ) ).

% Max.boundedE
tff(fact_6854_Max_OboundedE,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)),X3))
         => ! [A10: nat] :
              ( pp(aa_set_nat_bool(member_nat(A10),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A10),X3)) ) ) ) ) ).

% Max.boundedE
tff(fact_6855_Max_OboundedE,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(lattic8263393255366662781ax_int,A2)),X3))
         => ! [A10: int] :
              ( pp(aa_set_int_bool(member_int(A10),A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A10),X3)) ) ) ) ) ).

% Max.boundedE
tff(fact_6856_eq__Max__iff,axiom,
    ! [A2: set_Code_integer,M: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ( M = lattic4901227151466704046nteger(A2) )
        <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(M),A2))
            & ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A2))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X4),M)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6857_eq__Max__iff,axiom,
    ! [A2: set_real,M: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ( M = lattic4275903605611617917x_real(A2) )
        <=> ( pp(aa_set_real_bool(member_real(M),A2))
            & ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),M)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6858_eq__Max__iff,axiom,
    ! [A2: set_bool,M: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( ( pp(M)
          <=> pp(aa_set_bool_bool(lattic7308977977949639686x_bool,A2)) )
        <=> ( pp(aa_set_bool_bool(member_bool(M),A2))
            & ! [X4: bool] :
                ( pp(aa_set_bool_bool(member_bool(X4),A2))
               => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X4),M)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6859_eq__Max__iff,axiom,
    ! [A2: set_rat,M: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ( ( M = aa_set_rat_rat(lattic7630753665789217321ax_rat,A2) )
        <=> ( pp(aa_set_rat_bool(member_rat(M),A2))
            & ! [X4: rat] :
                ( pp(aa_set_rat_bool(member_rat(X4),A2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),M)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6860_eq__Max__iff,axiom,
    ! [A2: set_num,M: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ( ( M = aa_set_num_num(lattic4823215512031491691ax_num,A2) )
        <=> ( pp(aa_set_num_bool(member_num(M),A2))
            & ! [X4: num] :
                ( pp(aa_set_num_bool(member_num(X4),A2))
               => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),M)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6861_eq__Max__iff,axiom,
    ! [A2: set_nat,M: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( ( M = aa_set_nat_nat(lattic8265883725875713057ax_nat,A2) )
        <=> ( pp(aa_set_nat_bool(member_nat(M),A2))
            & ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),A2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6862_eq__Max__iff,axiom,
    ! [A2: set_int,M: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ( M = aa_set_int_int(lattic8263393255366662781ax_int,A2) )
        <=> ( pp(aa_set_int_bool(member_int(M),A2))
            & ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),A2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),M)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6863_Max__ge__iff,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X3),lattic4901227151466704046nteger(A2)))
        <=> ? [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A2))
              & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X3),X4)) ) ) ) ) ).

% Max_ge_iff
tff(fact_6864_Max__ge__iff,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),lattic4275903605611617917x_real(A2)))
        <=> ? [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),X4)) ) ) ) ) ).

% Max_ge_iff
tff(fact_6865_Max__ge__iff,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X3),aa_set_bool_bool(lattic7308977977949639686x_bool,A2)))
        <=> ? [X4: bool] :
              ( pp(aa_set_bool_bool(member_bool(X4),A2))
              & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X3),X4)) ) ) ) ) ).

% Max_ge_iff
tff(fact_6866_Max__ge__iff,axiom,
    ! [A2: set_rat,X3: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),aa_set_rat_rat(lattic7630753665789217321ax_rat,A2)))
        <=> ? [X4: rat] :
              ( pp(aa_set_rat_bool(member_rat(X4),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),X4)) ) ) ) ) ).

% Max_ge_iff
tff(fact_6867_Max__ge__iff,axiom,
    ! [A2: set_num,X3: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),aa_set_num_num(lattic4823215512031491691ax_num,A2)))
        <=> ? [X4: num] :
              ( pp(aa_set_num_bool(member_num(X4),A2))
              & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),X4)) ) ) ) ) ).

% Max_ge_iff
tff(fact_6868_Max__ge__iff,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)))
        <=> ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),X4)) ) ) ) ) ).

% Max_ge_iff
tff(fact_6869_Max__ge__iff,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_set_int_int(lattic8263393255366662781ax_int,A2)))
        <=> ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),X4)) ) ) ) ) ).

% Max_ge_iff
tff(fact_6870_Max__eq__iff,axiom,
    ! [A2: set_Code_integer,M: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ( lattic4901227151466704046nteger(A2) = M )
        <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(M),A2))
            & ! [X4: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A2))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X4),M)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6871_Max__eq__iff,axiom,
    ! [A2: set_real,M: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ( lattic4275903605611617917x_real(A2) = M )
        <=> ( pp(aa_set_real_bool(member_real(M),A2))
            & ! [X4: real] :
                ( pp(aa_set_real_bool(member_real(X4),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),M)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6872_Max__eq__iff,axiom,
    ! [A2: set_bool,M: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( ( pp(aa_set_bool_bool(lattic7308977977949639686x_bool,A2))
          <=> pp(M) )
        <=> ( pp(aa_set_bool_bool(member_bool(M),A2))
            & ! [X4: bool] :
                ( pp(aa_set_bool_bool(member_bool(X4),A2))
               => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X4),M)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6873_Max__eq__iff,axiom,
    ! [A2: set_rat,M: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ( ( aa_set_rat_rat(lattic7630753665789217321ax_rat,A2) = M )
        <=> ( pp(aa_set_rat_bool(member_rat(M),A2))
            & ! [X4: rat] :
                ( pp(aa_set_rat_bool(member_rat(X4),A2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),M)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6874_Max__eq__iff,axiom,
    ! [A2: set_num,M: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ( ( aa_set_num_num(lattic4823215512031491691ax_num,A2) = M )
        <=> ( pp(aa_set_num_bool(member_num(M),A2))
            & ! [X4: num] :
                ( pp(aa_set_num_bool(member_num(X4),A2))
               => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),M)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6875_Max__eq__iff,axiom,
    ! [A2: set_nat,M: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A2) = M )
        <=> ( pp(aa_set_nat_bool(member_nat(M),A2))
            & ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),A2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6876_Max__eq__iff,axiom,
    ! [A2: set_int,M: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ( aa_set_int_int(lattic8263393255366662781ax_int,A2) = M )
        <=> ( pp(aa_set_int_bool(member_int(M),A2))
            & ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),A2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),M)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6877_Max__gr__iff,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),lattic4901227151466704046nteger(A2)))
        <=> ? [X4: code_integer] :
              ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),A2))
              & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),X4)) ) ) ) ) ).

% Max_gr_iff
tff(fact_6878_Max__gr__iff,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X3),aa_set_bool_bool(lattic7308977977949639686x_bool,A2)))
        <=> ? [X4: bool] :
              ( pp(aa_set_bool_bool(member_bool(X4),A2))
              & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_bool,X3),X4)) ) ) ) ) ).

% Max_gr_iff
tff(fact_6879_Max__gr__iff,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),lattic4275903605611617917x_real(A2)))
        <=> ? [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),X4)) ) ) ) ) ).

% Max_gr_iff
tff(fact_6880_Max__gr__iff,axiom,
    ! [A2: set_rat,X3: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ( A2 != bot_bot_set_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),aa_set_rat_rat(lattic7630753665789217321ax_rat,A2)))
        <=> ? [X4: rat] :
              ( pp(aa_set_rat_bool(member_rat(X4),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),X4)) ) ) ) ) ).

% Max_gr_iff
tff(fact_6881_Max__gr__iff,axiom,
    ! [A2: set_num,X3: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ( A2 != bot_bot_set_num )
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),aa_set_num_num(lattic4823215512031491691ax_num,A2)))
        <=> ? [X4: num] :
              ( pp(aa_set_num_bool(member_num(X4),A2))
              & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),X4)) ) ) ) ) ).

% Max_gr_iff
tff(fact_6882_Max__gr__iff,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)))
        <=> ? [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X4)) ) ) ) ) ).

% Max_gr_iff
tff(fact_6883_Max__gr__iff,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),aa_set_int_int(lattic8263393255366662781ax_int,A2)))
        <=> ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),X4)) ) ) ) ) ).

% Max_gr_iff
tff(fact_6884_Max__insert2,axiom,
    ! [A2: set_bool,A: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ! [B2: bool] :
            ( pp(aa_set_bool_bool(member_bool(B2),A2))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,B2),A)) )
       => ( pp(aa_set_bool_bool(lattic7308977977949639686x_bool,aa_set_bool_set_bool(insert_bool(A),A2)))
        <=> pp(A) ) ) ) ).

% Max_insert2
tff(fact_6885_Max__insert2,axiom,
    ! [A2: set_real,A: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [B2: real] :
            ( pp(aa_set_real_bool(member_real(B2),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),A)) )
       => ( lattic4275903605611617917x_real(aa_set_real_set_real(insert_real(A),A2)) = A ) ) ) ).

% Max_insert2
tff(fact_6886_Max__insert2,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ! [B2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(B2),A2))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A)) )
       => ( lattic4901227151466704046nteger(insert_Code_integer(A,A2)) = A ) ) ) ).

% Max_insert2
tff(fact_6887_Max__insert2,axiom,
    ! [A2: set_rat,A: rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( ! [B2: rat] :
            ( pp(aa_set_rat_bool(member_rat(B2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B2),A)) )
       => ( aa_set_rat_rat(lattic7630753665789217321ax_rat,insert_rat(A,A2)) = A ) ) ) ).

% Max_insert2
tff(fact_6888_Max__insert2,axiom,
    ! [A2: set_num,A: num] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( ! [B2: num] :
            ( pp(aa_set_num_bool(member_num(B2),A2))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A)) )
       => ( aa_set_num_num(lattic4823215512031491691ax_num,insert_num(A,A2)) = A ) ) ) ).

% Max_insert2
tff(fact_6889_Max__insert2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [B2: nat] :
            ( pp(aa_set_nat_bool(member_nat(B2),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A)) )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(A),A2)) = A ) ) ) ).

% Max_insert2
tff(fact_6890_Max__insert2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [B2: int] :
            ( pp(aa_set_int_bool(member_int(B2),A2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A)) )
       => ( aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(insert_int(A),A2)) = A ) ) ) ).

% Max_insert2
tff(fact_6891_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_6892_real__add__less__0__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(uminus_uminus_real,X3))) ) ).

% real_add_less_0_iff
tff(fact_6893_real__0__less__add__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X3)),Y)) ) ).

% real_0_less_add_iff
tff(fact_6894_real__0__le__add__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X3)),Y)) ) ).

% real_0_le_add_iff
tff(fact_6895_real__add__le__0__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,X3))) ) ).

% real_add_le_0_iff
tff(fact_6896_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_6897_abs__real__def,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_real_def
tff(fact_6898_negative__zle__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int)) ).

% negative_zle_0
tff(fact_6899_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ~ ! [N2: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nonpos_int_cases
tff(fact_6900_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = I ) ) ) ).

% zabs_def
tff(fact_6901_finite__subset__Union,axiom,
    ! [A2: set_complex,B8: set_set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),comple8424636186594484919omplex(B8)))
       => ~ ! [F8: set_set_complex] :
              ( finite6551019134538273531omplex(F8)
             => ( ord_le4750530260501030778omplex(F8,B8)
               => ~ pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),comple8424636186594484919omplex(F8))) ) ) ) ) ).

% finite_subset_Union
tff(fact_6902_finite__subset__Union,axiom,
    ! [A2: set_Code_integer,B8: set_set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A2),comple739944243200306918nteger(B8)))
       => ~ ! [F8: set_set_Code_integer] :
              ( finite6931041176100689706nteger(F8)
             => ( ord_le1914454125413604393nteger(F8,B8)
               => ~ pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A2),comple739944243200306918nteger(F8))) ) ) ) ) ).

% finite_subset_Union
tff(fact_6903_finite__subset__Union,axiom,
    ! [A2: set_int,B8: set_set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),comple3221217463730067765et_int(B8)))
       => ~ ! [F8: set_set_int] :
              ( finite6197958912794628473et_int(F8)
             => ( ord_le4403425263959731960et_int(F8,B8)
               => ~ pp(aa_set_int_bool(ord_less_eq_set_int(A2),comple3221217463730067765et_int(F8))) ) ) ) ) ).

% finite_subset_Union
tff(fact_6904_finite__subset__Union,axiom,
    ! [A2: set_nat,B8: set_set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),comple7399068483239264473et_nat(B8)))
       => ~ ! [F8: set_set_nat] :
              ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,F8))
             => ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(F8),B8))
               => ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),comple7399068483239264473et_nat(F8))) ) ) ) ) ).

% finite_subset_Union
tff(fact_6905_cSup__asclose,axiom,
    ! [S3: set_int,L: int,E2: int] :
      ( ( S3 != bot_bot_set_int )
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),S3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),L))),E2)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_int_int(complete_Sup_Sup_int,S3)),L))),E2)) ) ) ).

% cSup_asclose
tff(fact_6906_cSup__asclose,axiom,
    ! [S3: set_real,L: real,E2: real] :
      ( ( S3 != bot_bot_set_real )
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),S3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),L))),E2)) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(comple1385675409528146559p_real(S3)),L))),E2)) ) ) ).

% cSup_asclose
tff(fact_6907_Sup__fold__sup,axiom,
    ! [A2: set_set_real] :
      ( finite9007344921179782393t_real(A2)
     => ( comple3096694443085538997t_real(A2) = finite8257685287532064654t_real(sup_sup_set_real,bot_bot_set_real,A2) ) ) ).

% Sup_fold_sup
tff(fact_6908_Sup__fold__sup,axiom,
    ! [A2: set_set_bool] :
      ( finite3197276090331229984t_bool(A2)
     => ( comple9167723193526752868t_bool(A2) = finite1372036876905702536t_bool(sup_sup_set_bool,bot_bot_set_bool,A2) ) ) ).

% Sup_fold_sup
tff(fact_6909_Sup__fold__sup,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( comple3221217463730067765et_int(A2) = finite2508883933289561102et_int(sup_sup_set_int,bot_bot_set_int,A2) ) ) ).

% Sup_fold_sup
tff(fact_6910_Sup__fold__sup,axiom,
    ! [A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(finite1152437895449049373et_nat,A2))
     => ( comple7399068483239264473et_nat(A2) = finite677925301803182934et_nat(sup_sup_set_nat,bot_bot_set_nat,A2) ) ) ).

% Sup_fold_sup
tff(fact_6911_Max_Osubset__imp,axiom,
    ! [A2: set_Code_integer,B3: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(A2),B3))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A2)),lattic4901227151466704046nteger(B3))) ) ) ) ).

% Max.subset_imp
tff(fact_6912_Max_Osubset__imp,axiom,
    ! [A2: set_real,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
     => ( ( A2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(finite_finite_real,B3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,lattic4275903605611617917x_real(A2)),lattic4275903605611617917x_real(B3))) ) ) ) ).

% Max.subset_imp
tff(fact_6913_Max_Osubset__imp,axiom,
    ! [A2: set_bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(A2),B3))
     => ( ( A2 != bot_bot_set_bool )
       => ( pp(aa_set_bool_bool(finite_finite_bool,B3))
         => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_set_bool_bool(lattic7308977977949639686x_bool,A2)),aa_set_bool_bool(lattic7308977977949639686x_bool,B3))) ) ) ) ).

% Max.subset_imp
tff(fact_6914_Max_Osubset__imp,axiom,
    ! [A2: set_rat,B3: set_rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(A2),B3))
     => ( ( A2 != bot_bot_set_rat )
       => ( pp(aa_set_rat_bool(finite_finite_rat,B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,A2)),aa_set_rat_rat(lattic7630753665789217321ax_rat,B3))) ) ) ) ).

% Max.subset_imp
tff(fact_6915_Max_Osubset__imp,axiom,
    ! [A2: set_num,B3: set_num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(A2),B3))
     => ( ( A2 != bot_bot_set_num )
       => ( pp(aa_set_num_bool(finite_finite_num,B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_set_num_num(lattic4823215512031491691ax_num,A2)),aa_set_num_num(lattic4823215512031491691ax_num,B3))) ) ) ) ).

% Max.subset_imp
tff(fact_6916_Max_Osubset__imp,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( ( A2 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)),aa_set_nat_nat(lattic8265883725875713057ax_nat,B3))) ) ) ) ).

% Max.subset_imp
tff(fact_6917_Max_Osubset__imp,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( ( A2 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(finite_finite_int,B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(lattic8263393255366662781ax_int,A2)),aa_set_int_int(lattic8263393255366662781ax_int,B3))) ) ) ) ).

% Max.subset_imp
tff(fact_6918_Max__mono,axiom,
    ! [M8: set_Code_integer,N7: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(M8),N7))
     => ( ( M8 != bot_bo3990330152332043303nteger )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,N7))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(M8)),lattic4901227151466704046nteger(N7))) ) ) ) ).

% Max_mono
tff(fact_6919_Max__mono,axiom,
    ! [M8: set_real,N7: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(M8),N7))
     => ( ( M8 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(finite_finite_real,N7))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,lattic4275903605611617917x_real(M8)),lattic4275903605611617917x_real(N7))) ) ) ) ).

% Max_mono
tff(fact_6920_Max__mono,axiom,
    ! [M8: set_bool,N7: set_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(M8),N7))
     => ( ( M8 != bot_bot_set_bool )
       => ( pp(aa_set_bool_bool(finite_finite_bool,N7))
         => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_set_bool_bool(lattic7308977977949639686x_bool,M8)),aa_set_bool_bool(lattic7308977977949639686x_bool,N7))) ) ) ) ).

% Max_mono
tff(fact_6921_Max__mono,axiom,
    ! [M8: set_rat,N7: set_rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(M8),N7))
     => ( ( M8 != bot_bot_set_rat )
       => ( pp(aa_set_rat_bool(finite_finite_rat,N7))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,M8)),aa_set_rat_rat(lattic7630753665789217321ax_rat,N7))) ) ) ) ).

% Max_mono
tff(fact_6922_Max__mono,axiom,
    ! [M8: set_num,N7: set_num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(M8),N7))
     => ( ( M8 != bot_bot_set_num )
       => ( pp(aa_set_num_bool(finite_finite_num,N7))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_set_num_num(lattic4823215512031491691ax_num,M8)),aa_set_num_num(lattic4823215512031491691ax_num,N7))) ) ) ) ).

% Max_mono
tff(fact_6923_Max__mono,axiom,
    ! [M8: set_nat,N7: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,M8),N7))
     => ( ( M8 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(finite_finite_nat,N7))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,M8)),aa_set_nat_nat(lattic8265883725875713057ax_nat,N7))) ) ) ) ).

% Max_mono
tff(fact_6924_Max__mono,axiom,
    ! [M8: set_int,N7: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(M8),N7))
     => ( ( M8 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(finite_finite_int,N7))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_set_int_int(lattic8263393255366662781ax_int,M8)),aa_set_int_int(lattic8263393255366662781ax_int,N7))) ) ) ) ).

% Max_mono
tff(fact_6925_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ) ).

% int_cases3
tff(fact_6926_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))) ).

% not_zle_0_negative
tff(fact_6927_negative__zless__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int)) ).

% negative_zless_0
tff(fact_6928_negD,axiom,
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),zero_zero_int))
     => ? [N2: nat] : X3 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_6929_foldl__set,axiom,
    ! [L: list_set_real] : foldl_4142838678319929319t_real(sup_sup_set_real,bot_bot_set_real,L) = comple3096694443085538997t_real(collect_set_real(aTP_Lamm_xp(L))) ).

% foldl_set
tff(fact_6930_foldl__set,axiom,
    ! [L: list_set_bool] : foldl_3150261563131881825t_bool(sup_sup_set_bool,bot_bot_set_bool,L) = comple9167723193526752868t_bool(collect_set_bool(aTP_Lamm_xq(L))) ).

% foldl_set
tff(fact_6931_foldl__set,axiom,
    ! [L: list_set_int] : foldl_6819690284573351271et_int(sup_sup_set_int,bot_bot_set_int,L) = comple3221217463730067765et_int(collect_set_int(aTP_Lamm_xr(L))) ).

% foldl_set
tff(fact_6932_foldl__set,axiom,
    ! [L: list_set_nat] : foldl_4988731653086973103et_nat(sup_sup_set_nat,bot_bot_set_nat,L) = comple7399068483239264473et_nat(collect_set_nat(aTP_Lamm_xs(L))) ).

% foldl_set
tff(fact_6933_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_6934_sgn__real__def,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_real_def
tff(fact_6935_Max__add__commute,axiom,
    ! [S3: set_complex,F: fun_complex_real,K: real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ( lattic4275903605611617917x_real(image_complex_real(aa_rea4562580341872191937x_real(aTP_Lamm_xt(F),K),S3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic4275903605611617917x_real(image_complex_real(F,S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6936_Max__add__commute,axiom,
    ! [S3: set_Code_integer,F: fun_Co3960896333048504454r_real,K: real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( lattic4275903605611617917x_real(image_7738145705984076560r_real(aa_rea8190550742337927152r_real(aTP_Lamm_xu(F),K),S3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic4275903605611617917x_real(image_7738145705984076560r_real(F,S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6937_Max__add__commute,axiom,
    ! [S3: set_complex,F: fun_complex_rat,K: rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ( aa_set_rat_rat(lattic7630753665789217321ax_rat,image_complex_rat(aa_rat15117248669498777ex_rat(aTP_Lamm_xv(F),K),S3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,image_complex_rat(F,S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6938_Max__add__commute,axiom,
    ! [S3: set_Code_integer,F: fun_Code_integer_rat,K: rat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( aa_set_rat_rat(lattic7630753665789217321ax_rat,image_315895873841295420er_rat(aa_rat6553461586499541832er_rat(aTP_Lamm_xw(F),K),S3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,image_315895873841295420er_rat(F,S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6939_Max__add__commute,axiom,
    ! [S3: set_real,F: fun_real_real,K: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( S3 != bot_bot_set_real )
       => ( lattic4275903605611617917x_real(image_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_xx(F),K),S3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic4275903605611617917x_real(image_real_real(F,S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6940_Max__add__commute,axiom,
    ! [S3: set_real,F: fun_real_rat,K: rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( S3 != bot_bot_set_real )
       => ( aa_set_rat_rat(lattic7630753665789217321ax_rat,image_real_rat(aa_rat_fun_real_rat(aTP_Lamm_xy(F),K),S3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,image_real_rat(F,S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6941_Max__add__commute,axiom,
    ! [S3: set_bool,F: fun_bool_real,K: real] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ( ( S3 != bot_bot_set_bool )
       => ( lattic4275903605611617917x_real(image_bool_real(aa_rea8194757448443630444l_real(aTP_Lamm_xz(F),K),S3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic4275903605611617917x_real(image_bool_real(F,S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6942_Max__add__commute,axiom,
    ! [S3: set_bool,F: fun_bool_rat,K: rat] :
      ( pp(aa_set_bool_bool(finite_finite_bool,S3))
     => ( ( S3 != bot_bot_set_bool )
       => ( aa_set_rat_rat(lattic7630753665789217321ax_rat,image_bool_rat(aa_rat_fun_bool_rat(aTP_Lamm_ya(F),K),S3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,image_bool_rat(F,S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6943_Max__add__commute,axiom,
    ! [S3: set_nat,F: fun_nat_real,K: real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ( lattic4275903605611617917x_real(aa_set_nat_set_real(image_nat_real(aa_real_fun_nat_real(aTP_Lamm_yb(F),K)),S3)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic4275903605611617917x_real(aa_set_nat_set_real(image_nat_real(F),S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6944_Max__add__commute,axiom,
    ! [S3: set_nat,F: fun_nat_rat,K: rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ( aa_set_rat_rat(lattic7630753665789217321ax_rat,image_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_yc(F),K),S3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_rat_rat(lattic7630753665789217321ax_rat,image_nat_rat(F,S3))),K) ) ) ) ).

% Max_add_commute
tff(fact_6945_Id__on__def,axiom,
    ! [A2: set_nat] : id_on_nat(A2) = comple5685304695842803022at_nat(image_7178329752028323786at_nat(aTP_Lamm_yd,A2)) ).

% Id_on_def
tff(fact_6946_Id__on__def,axiom,
    ! [A2: set_int] : id_on_int(A2) = comple5382143125604098054nt_int(image_8635204845542730022nt_int(aTP_Lamm_ye,A2)) ).

% Id_on_def
tff(fact_6947_Id__on__def,axiom,
    ! [A2: set_Code_integer] : id_on_Code_integer(A2) = comple2203973573673791208nteger(image_8529064383210301591nteger(aTP_Lamm_yf,A2)) ).

% Id_on_def
tff(fact_6948_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% neg_int_cases
tff(fact_6949_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)))
     => ( ( B != zero_zero_int )
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)),aa_int_int(uminus_uminus_int,Q4)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q4)),one_one_int))),aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)),zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2))))) ) ) ).

% zminus1_lemma
tff(fact_6950_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Summary),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_6951_UN__insert,axiom,
    ! [B3: fun_nat_set_nat,A: nat,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),aa_set_nat_set_nat(insert_nat(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(B3,A)),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2))) ).

% UN_insert
tff(fact_6952_UN__insert,axiom,
    ! [B3: fun_VE9028255658833972685et_nat,A: vEBT_VEBT,A2: set_VEBT_VEBT] : comple7399068483239264473et_nat(image_6451421511446451829et_nat(B3,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_VEBT_VEBT_set_nat(B3,A)),comple7399068483239264473et_nat(image_6451421511446451829et_nat(B3,A2))) ).

% UN_insert
tff(fact_6953_UN__insert,axiom,
    ! [B3: fun_int_set_nat,A: int,A2: set_int] : comple7399068483239264473et_nat(image_int_set_nat(B3,aa_set_int_set_int(insert_int(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_int_set_nat(B3,A)),comple7399068483239264473et_nat(image_int_set_nat(B3,A2))) ).

% UN_insert
tff(fact_6954_UN__insert,axiom,
    ! [B3: fun_bool_set_nat,A: bool,A2: set_bool] : comple7399068483239264473et_nat(image_bool_set_nat(B3,aa_set_bool_set_bool(insert_bool(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_bool_set_nat(B3,A)),comple7399068483239264473et_nat(image_bool_set_nat(B3,A2))) ).

% UN_insert
tff(fact_6955_UN__insert,axiom,
    ! [B3: fun_real_set_nat,A: real,A2: set_real] : comple7399068483239264473et_nat(image_real_set_nat(B3,aa_set_real_set_real(insert_real(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_real_set_nat(B3,A)),comple7399068483239264473et_nat(image_real_set_nat(B3,A2))) ).

% UN_insert
tff(fact_6956_UN__simps_I2_J,axiom,
    ! [C2: set_real,A2: fun_real_set_real,B3: set_real] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aa_set2944350243680128683t_real(aTP_Lamm_yg(A2),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aa_set2944350243680128683t_real(aTP_Lamm_yg(A2),B3),C2)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,comple3096694443085538997t_real(image_real_set_real(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6957_UN__simps_I2_J,axiom,
    ! [C2: set_real,A2: fun_real_set_bool,B3: set_bool] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aa_set5813937368821140009t_bool(aTP_Lamm_yh(A2),B3),C2)) = bot_bot_set_bool ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aa_set5813937368821140009t_bool(aTP_Lamm_yh(A2),B3),C2)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,comple9167723193526752868t_bool(image_real_set_bool(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6958_UN__simps_I2_J,axiom,
    ! [C2: set_real,A2: fun_real_set_int,B3: set_int] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aa_set4051449307867115051et_int(aTP_Lamm_yi(A2),B3),C2)) = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aa_set4051449307867115051et_int(aTP_Lamm_yi(A2),B3),C2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,comple3221217463730067765et_int(image_real_set_int(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6959_UN__simps_I2_J,axiom,
    ! [C2: set_bool,A2: fun_bool_set_real,B3: set_real] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aa_set1647705334557766892t_real(aTP_Lamm_yj(A2),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aa_set1647705334557766892t_real(aTP_Lamm_yj(A2),B3),C2)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,comple3096694443085538997t_real(image_bool_set_real(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6960_UN__simps_I2_J,axiom,
    ! [C2: set_bool,A2: fun_bool_set_bool,B3: set_bool] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aa_set942686091973210910t_bool(aTP_Lamm_yk(A2),B3),C2)) = bot_bot_set_bool ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aa_set942686091973210910t_bool(aTP_Lamm_yk(A2),B3),C2)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,comple9167723193526752868t_bool(image_bool_set_bool(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6961_UN__simps_I2_J,axiom,
    ! [C2: set_bool,A2: fun_bool_set_int,B3: set_int] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aa_set1195314601623344620et_int(aTP_Lamm_yl(A2),B3),C2)) = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aa_set1195314601623344620et_int(aTP_Lamm_yl(A2),B3),C2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,comple3221217463730067765et_int(image_bool_set_int(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6962_UN__simps_I2_J,axiom,
    ! [C2: set_nat,A2: fun_nat_set_real,B3: set_real] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aa_set5039813165413123151t_real(aTP_Lamm_ym(A2),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aa_set5039813165413123151t_real(aTP_Lamm_ym(A2),B3),C2)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,comple3096694443085538997t_real(image_nat_set_real(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6963_UN__simps_I2_J,axiom,
    ! [C2: set_nat,A2: fun_nat_set_bool,B3: set_bool] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( comple9167723193526752868t_bool(image_nat_set_bool(aa_set6340204957045714437t_bool(aTP_Lamm_yn(A2),B3),C2)) = bot_bot_set_bool ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( comple9167723193526752868t_bool(image_nat_set_bool(aa_set6340204957045714437t_bool(aTP_Lamm_yn(A2),B3),C2)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,comple9167723193526752868t_bool(image_nat_set_bool(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6964_UN__simps_I2_J,axiom,
    ! [C2: set_nat,A2: fun_nat_set_int,B3: set_int] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( comple3221217463730067765et_int(image_nat_set_int(aa_set2464685511577892303et_int(aTP_Lamm_yo(A2),B3),C2)) = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( comple3221217463730067765et_int(image_nat_set_int(aa_set2464685511577892303et_int(aTP_Lamm_yo(A2),B3),C2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,comple3221217463730067765et_int(image_nat_set_int(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6965_UN__simps_I2_J,axiom,
    ! [C2: set_int,A2: fun_int_set_real,B3: set_real] :
      ( ( ( C2 = bot_bot_set_int )
       => ( comple3096694443085538997t_real(image_int_set_real(aa_set1825250779553922219t_real(aTP_Lamm_yp(A2),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_int )
       => ( comple3096694443085538997t_real(image_int_set_real(aa_set1825250779553922219t_real(aTP_Lamm_yp(A2),B3),C2)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,comple3096694443085538997t_real(image_int_set_real(A2,C2))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_6966_UN__simps_I3_J,axiom,
    ! [C2: set_real,A2: set_real,B3: fun_real_set_real] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aa_fun686715674179075668t_real(aTP_Lamm_yq(A2),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aa_fun686715674179075668t_real(aTP_Lamm_yq(A2),B3),C2)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),comple3096694443085538997t_real(image_real_set_real(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6967_UN__simps_I3_J,axiom,
    ! [C2: set_real,A2: set_bool,B3: fun_real_set_bool] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aa_fun7454438436055979324t_bool(aTP_Lamm_yr(A2),B3),C2)) = bot_bot_set_bool ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aa_fun7454438436055979324t_bool(aTP_Lamm_yr(A2),B3),C2)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),comple9167723193526752868t_bool(image_real_set_bool(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6968_UN__simps_I3_J,axiom,
    ! [C2: set_real,A2: set_int,B3: fun_real_set_int] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aa_fun7998497282572583252et_int(aTP_Lamm_ys(A2),B3),C2)) = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aa_fun7998497282572583252et_int(aTP_Lamm_ys(A2),B3),C2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),comple3221217463730067765et_int(image_real_set_int(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6969_UN__simps_I3_J,axiom,
    ! [C2: set_bool,A2: set_real,B3: fun_bool_set_real] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aa_fun418618139989480252t_real(aTP_Lamm_yt(A2),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aa_fun418618139989480252t_real(aTP_Lamm_yt(A2),B3),C2)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),comple3096694443085538997t_real(image_bool_set_real(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6970_UN__simps_I3_J,axiom,
    ! [C2: set_bool,A2: set_bool,B3: fun_bool_set_bool] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aa_fun2947227189488945156t_bool(aTP_Lamm_yu(A2),B3),C2)) = bot_bot_set_bool ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aa_fun2947227189488945156t_bool(aTP_Lamm_yu(A2),B3),C2)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),comple9167723193526752868t_bool(image_bool_set_bool(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6971_UN__simps_I3_J,axiom,
    ! [C2: set_bool,A2: set_int,B3: fun_bool_set_int] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aa_fun4225254755057748412et_int(aTP_Lamm_yv(A2),B3),C2)) = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aa_fun4225254755057748412et_int(aTP_Lamm_yv(A2),B3),C2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),comple3221217463730067765et_int(image_bool_set_int(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6972_UN__simps_I3_J,axiom,
    ! [C2: set_nat,A2: set_real,B3: fun_nat_set_real] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aa_fun7593858674118267548t_real(aTP_Lamm_yw(A2),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aa_fun7593858674118267548t_real(aTP_Lamm_yw(A2),B3),C2)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),comple3096694443085538997t_real(image_nat_set_real(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6973_UN__simps_I3_J,axiom,
    ! [C2: set_nat,A2: set_bool,B3: fun_nat_set_bool] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( comple9167723193526752868t_bool(image_nat_set_bool(aa_fun4930178584814763708t_bool(aTP_Lamm_yx(A2),B3),C2)) = bot_bot_set_bool ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( comple9167723193526752868t_bool(image_nat_set_bool(aa_fun4930178584814763708t_bool(aTP_Lamm_yx(A2),B3),C2)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),comple9167723193526752868t_bool(image_nat_set_bool(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6974_UN__simps_I3_J,axiom,
    ! [C2: set_nat,A2: set_int,B3: fun_nat_set_int] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( comple3221217463730067765et_int(image_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_yy(A2),B3),C2)) = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( comple3221217463730067765et_int(image_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_yy(A2),B3),C2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),comple3221217463730067765et_int(image_nat_set_int(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6975_UN__simps_I3_J,axiom,
    ! [C2: set_int,A2: set_real,B3: fun_int_set_real] :
      ( ( ( C2 = bot_bot_set_int )
       => ( comple3096694443085538997t_real(image_int_set_real(aa_fun1830857766034312532t_real(aTP_Lamm_yz(A2),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_int )
       => ( comple3096694443085538997t_real(image_int_set_real(aa_fun1830857766034312532t_real(aTP_Lamm_yz(A2),B3),C2)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),comple3096694443085538997t_real(image_int_set_real(B3,C2))) ) ) ) ).

% UN_simps(3)
tff(fact_6976_UN__simps_I1_J,axiom,
    ! [C2: set_real,A: vEBT_VEBT,B3: fun_re4523157989745374273T_VEBT] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(aa_fun7713332347920806140T_VEBT(aTP_Lamm_za(A),B3),C2)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(aa_fun7713332347920806140T_VEBT(aTP_Lamm_za(A),B3),C2)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6977_UN__simps_I1_J,axiom,
    ! [C2: set_real,A: real,B3: fun_real_set_real] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aa_fun686715674179075668t_real(aTP_Lamm_zb(A),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aa_fun686715674179075668t_real(aTP_Lamm_zb(A),B3),C2)) = aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_real_set_real(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6978_UN__simps_I1_J,axiom,
    ! [C2: set_real,A: bool,B3: fun_real_set_bool] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aa_fun7454438436055979324t_bool(aTP_Lamm_zc(A),B3),C2)) = bot_bot_set_bool ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aa_fun7454438436055979324t_bool(aTP_Lamm_zc(A),B3),C2)) = aa_set_bool_set_bool(insert_bool(A),comple9167723193526752868t_bool(image_real_set_bool(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6979_UN__simps_I1_J,axiom,
    ! [C2: set_real,A: int,B3: fun_real_set_int] :
      ( ( ( C2 = bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aa_fun7998497282572583252et_int(aTP_Lamm_zd(A),B3),C2)) = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aa_fun7998497282572583252et_int(aTP_Lamm_zd(A),B3),C2)) = aa_set_int_set_int(insert_int(A),comple3221217463730067765et_int(image_real_set_int(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6980_UN__simps_I1_J,axiom,
    ! [C2: set_bool,A: vEBT_VEBT,B3: fun_bo1266566265771860096T_VEBT] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple2820511241208326657T_VEBT(image_5756441063665793602T_VEBT(aa_fun8832029763029799358T_VEBT(aTP_Lamm_ze(A),B3),C2)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple2820511241208326657T_VEBT(image_5756441063665793602T_VEBT(aa_fun8832029763029799358T_VEBT(aTP_Lamm_ze(A),B3),C2)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_5756441063665793602T_VEBT(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6981_UN__simps_I1_J,axiom,
    ! [C2: set_bool,A: real,B3: fun_bool_set_real] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aa_fun418618139989480252t_real(aTP_Lamm_zf(A),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aa_fun418618139989480252t_real(aTP_Lamm_zf(A),B3),C2)) = aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_bool_set_real(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6982_UN__simps_I1_J,axiom,
    ! [C2: set_bool,A: bool,B3: fun_bool_set_bool] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aa_fun2947227189488945156t_bool(aTP_Lamm_zg(A),B3),C2)) = bot_bot_set_bool ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aa_fun2947227189488945156t_bool(aTP_Lamm_zg(A),B3),C2)) = aa_set_bool_set_bool(insert_bool(A),comple9167723193526752868t_bool(image_bool_set_bool(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6983_UN__simps_I1_J,axiom,
    ! [C2: set_bool,A: int,B3: fun_bool_set_int] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aa_fun4225254755057748412et_int(aTP_Lamm_zh(A),B3),C2)) = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aa_fun4225254755057748412et_int(aTP_Lamm_zh(A),B3),C2)) = aa_set_int_set_int(insert_int(A),comple3221217463730067765et_int(image_bool_set_int(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6984_UN__simps_I1_J,axiom,
    ! [C2: set_nat,A: vEBT_VEBT,B3: fun_na7453361831206303133T_VEBT] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(aa_fun7681330223414995964T_VEBT(aTP_Lamm_zi(A),B3),C2)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(aa_fun7681330223414995964T_VEBT(aTP_Lamm_zi(A),B3),C2)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6985_UN__simps_I1_J,axiom,
    ! [C2: set_nat,A: real,B3: fun_nat_set_real] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aa_fun7593858674118267548t_real(aTP_Lamm_zj(A),B3),C2)) = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aa_fun7593858674118267548t_real(aTP_Lamm_zj(A),B3),C2)) = aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_nat_set_real(B3,C2))) ) ) ) ).

% UN_simps(1)
tff(fact_6986_UN__singleton,axiom,
    ! [A2: set_VEBT_VEBT] : comple2820511241208326657T_VEBT(image_2685870239581809509T_VEBT(aTP_Lamm_zk,A2)) = A2 ).

% UN_singleton
tff(fact_6987_UN__singleton,axiom,
    ! [A2: set_real] : comple3096694443085538997t_real(image_real_set_real(aTP_Lamm_zl,A2)) = A2 ).

% UN_singleton
tff(fact_6988_UN__singleton,axiom,
    ! [A2: set_bool] : comple9167723193526752868t_bool(image_bool_set_bool(aTP_Lamm_zm,A2)) = A2 ).

% UN_singleton
tff(fact_6989_UN__singleton,axiom,
    ! [A2: set_int] : comple3221217463730067765et_int(image_int_set_int(aTP_Lamm_zn,A2)) = A2 ).

% UN_singleton
tff(fact_6990_UN__singleton,axiom,
    ! [A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_zo),A2)) = A2 ).

% UN_singleton
tff(fact_6991_ccSUP__empty,axiom,
    ! [F: fun_real_set_real] : comple3096694443085538997t_real(image_real_set_real(F,bot_bot_set_real)) = bot_bot_set_real ).

% ccSUP_empty
tff(fact_6992_ccSUP__empty,axiom,
    ! [F: fun_real_set_bool] : comple9167723193526752868t_bool(image_real_set_bool(F,bot_bot_set_real)) = bot_bot_set_bool ).

% ccSUP_empty
tff(fact_6993_ccSUP__empty,axiom,
    ! [F: fun_real_set_int] : comple3221217463730067765et_int(image_real_set_int(F,bot_bot_set_real)) = bot_bot_set_int ).

% ccSUP_empty
tff(fact_6994_ccSUP__empty,axiom,
    ! [F: fun_bool_set_real] : comple3096694443085538997t_real(image_bool_set_real(F,bot_bot_set_bool)) = bot_bot_set_real ).

% ccSUP_empty
tff(fact_6995_ccSUP__empty,axiom,
    ! [F: fun_bool_set_bool] : comple9167723193526752868t_bool(image_bool_set_bool(F,bot_bot_set_bool)) = bot_bot_set_bool ).

% ccSUP_empty
tff(fact_6996_ccSUP__empty,axiom,
    ! [F: fun_bool_set_int] : comple3221217463730067765et_int(image_bool_set_int(F,bot_bot_set_bool)) = bot_bot_set_int ).

% ccSUP_empty
tff(fact_6997_ccSUP__empty,axiom,
    ! [F: fun_nat_set_real] : comple3096694443085538997t_real(image_nat_set_real(F,bot_bot_set_nat)) = bot_bot_set_real ).

% ccSUP_empty
tff(fact_6998_ccSUP__empty,axiom,
    ! [F: fun_nat_set_bool] : comple9167723193526752868t_bool(image_nat_set_bool(F,bot_bot_set_nat)) = bot_bot_set_bool ).

% ccSUP_empty
tff(fact_6999_ccSUP__empty,axiom,
    ! [F: fun_nat_set_int] : comple3221217463730067765et_int(image_nat_set_int(F,bot_bot_set_nat)) = bot_bot_set_int ).

% ccSUP_empty
tff(fact_7000_ccSUP__empty,axiom,
    ! [F: fun_int_set_real] : comple3096694443085538997t_real(image_int_set_real(F,bot_bot_set_int)) = bot_bot_set_real ).

% ccSUP_empty
tff(fact_7001_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_set_real] :
      ( ( comple3096694443085538997t_real(A2) = bot_bot_set_real )
    <=> ! [X4: set_real] :
          ( member_set_real(X4,A2)
         => ( X4 = bot_bot_set_real ) ) ) ).

% Sup_bot_conv(1)
tff(fact_7002_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_set_bool] :
      ( ( comple9167723193526752868t_bool(A2) = bot_bot_set_bool )
    <=> ! [X4: set_bool] :
          ( member_set_bool(X4,A2)
         => ( X4 = bot_bot_set_bool ) ) ) ).

% Sup_bot_conv(1)
tff(fact_7003_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_set_int] :
      ( ( comple3221217463730067765et_int(A2) = bot_bot_set_int )
    <=> ! [X4: set_int] :
          ( member_set_int(X4,A2)
         => ( X4 = bot_bot_set_int ) ) ) ).

% Sup_bot_conv(1)
tff(fact_7004_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_set_nat] :
      ( ( comple7399068483239264473et_nat(A2) = bot_bot_set_nat )
    <=> ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => ( X4 = bot_bot_set_nat ) ) ) ).

% Sup_bot_conv(1)
tff(fact_7005_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_set_real] :
      ( ( bot_bot_set_real = comple3096694443085538997t_real(A2) )
    <=> ! [X4: set_real] :
          ( member_set_real(X4,A2)
         => ( X4 = bot_bot_set_real ) ) ) ).

% Sup_bot_conv(2)
tff(fact_7006_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_set_bool] :
      ( ( bot_bot_set_bool = comple9167723193526752868t_bool(A2) )
    <=> ! [X4: set_bool] :
          ( member_set_bool(X4,A2)
         => ( X4 = bot_bot_set_bool ) ) ) ).

% Sup_bot_conv(2)
tff(fact_7007_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_set_int] :
      ( ( bot_bot_set_int = comple3221217463730067765et_int(A2) )
    <=> ! [X4: set_int] :
          ( member_set_int(X4,A2)
         => ( X4 = bot_bot_set_int ) ) ) ).

% Sup_bot_conv(2)
tff(fact_7008_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_set_nat] :
      ( ( bot_bot_set_nat = comple7399068483239264473et_nat(A2) )
    <=> ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => ( X4 = bot_bot_set_nat ) ) ) ).

% Sup_bot_conv(2)
tff(fact_7009_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_7010_SUP__identity__eq,axiom,
    ! [A2: set_int] : aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(aTP_Lamm_dm),A2)) = aa_set_int_int(complete_Sup_Sup_int,A2) ).

% SUP_identity_eq
tff(fact_7011_SUP__identity__eq,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_dl),A2)) = aa_set_nat_nat(complete_Sup_Sup_nat,A2) ).

% SUP_identity_eq
tff(fact_7012_SUP__identity__eq,axiom,
    ! [A2: set_set_nat] : comple7399068483239264473et_nat(image_7916887816326733075et_nat(aTP_Lamm_zp,A2)) = comple7399068483239264473et_nat(A2) ).

% SUP_identity_eq
tff(fact_7013_SUP__identity__eq,axiom,
    ! [A2: set_real] : comple1385675409528146559p_real(image_real_real(aTP_Lamm_dj,A2)) = comple1385675409528146559p_real(A2) ).

% SUP_identity_eq
tff(fact_7014_UN__I,axiom,
    ! [A: nat,A2: set_nat,B: vEBT_VEBT,B3: fun_na7453361831206303133T_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_nat_set_VEBT_VEBT(B3,A)))
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(B3,A2)))) ) ) ).

% UN_I
tff(fact_7015_UN__I,axiom,
    ! [A: nat,A2: set_nat,B: real,B3: fun_nat_set_real] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( pp(aa_set_real_bool(member_real(B),aa_nat_set_real(B3,A)))
       => pp(aa_set_real_bool(member_real(B),comple3096694443085538997t_real(image_nat_set_real(B3,A2)))) ) ) ).

% UN_I
tff(fact_7016_UN__I,axiom,
    ! [A: nat,A2: set_nat,B: int,B3: fun_nat_set_int] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( pp(aa_set_int_bool(member_int(B),aa_nat_set_int(B3,A)))
       => pp(aa_set_int_bool(member_int(B),comple3221217463730067765et_int(image_nat_set_int(B3,A2)))) ) ) ).

% UN_I
tff(fact_7017_UN__I,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,B3: fun_VE3640887651371881891T_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),A2))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_VEB8154608084236005916T_VEBT(B3,A)))
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),comple2820511241208326657T_VEBT(image_2685870239581809509T_VEBT(B3,A2)))) ) ) ).

% UN_I
tff(fact_7018_UN__I,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: real,B3: fun_VE2077312251927228713t_real] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),A2))
     => ( pp(aa_set_real_bool(member_real(B),aa_VEB5302157898331830618t_real(B3,A)))
       => pp(aa_set_real_bool(member_real(B),comple3096694443085538997t_real(image_6636839513470643793t_real(B3,A2)))) ) ) ).

% UN_I
tff(fact_7019_UN__I,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: int,B3: fun_VE6538172513902245929et_int] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),A2))
     => ( pp(aa_set_int_bool(member_int(B),aa_VEBT_VEBT_set_int(B3,A)))
       => pp(aa_set_int_bool(member_int(B),comple3221217463730067765et_int(image_2273570491937255121et_int(B3,A2)))) ) ) ).

% UN_I
tff(fact_7020_UN__I,axiom,
    ! [A: real,A2: set_real,B: vEBT_VEBT,B3: fun_re4523157989745374273T_VEBT] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_rea5591236203076396202T_VEBT(B3,A)))
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(B3,A2)))) ) ) ).

% UN_I
tff(fact_7021_UN__I,axiom,
    ! [A: real,A2: set_real,B: real,B3: fun_real_set_real] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( pp(aa_set_real_bool(member_real(B),aa_real_set_real(B3,A)))
       => pp(aa_set_real_bool(member_real(B),comple3096694443085538997t_real(image_real_set_real(B3,A2)))) ) ) ).

% UN_I
tff(fact_7022_UN__I,axiom,
    ! [A: real,A2: set_real,B: int,B3: fun_real_set_int] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( pp(aa_set_int_bool(member_int(B),aa_real_set_int(B3,A)))
       => pp(aa_set_int_bool(member_int(B),comple3221217463730067765et_int(image_real_set_int(B3,A2)))) ) ) ).

% UN_I
tff(fact_7023_UN__I,axiom,
    ! [A: int,A2: set_int,B: vEBT_VEBT,B3: fun_in7262655915974039105T_VEBT] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_int_set_VEBT_VEBT(B3,A)))
       => pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),comple2820511241208326657T_VEBT(image_1216245965459964705T_VEBT(B3,A2)))) ) ) ).

% UN_I
tff(fact_7024_UN__iff,axiom,
    ! [B: nat,B3: fun_nat_set_nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(B),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2))))
    <=> ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
          & pp(aa_set_nat_bool(member_nat(B),aa_nat_set_nat(B3,X4))) ) ) ).

% UN_iff
tff(fact_7025_Sup__empty,axiom,
    comple3096694443085538997t_real(bot_bot_set_set_real) = bot_bot_set_real ).

% Sup_empty
tff(fact_7026_Sup__empty,axiom,
    comple9167723193526752868t_bool(bot_bot_set_set_bool) = bot_bot_set_bool ).

% Sup_empty
tff(fact_7027_Sup__empty,axiom,
    comple3221217463730067765et_int(bot_bot_set_set_int) = bot_bot_set_int ).

% Sup_empty
tff(fact_7028_Sup__empty,axiom,
    ( pp(aa_set_bool_bool(comple6445916836919887748p_bool,bot_bot_set_bool))
  <=> bot_bot_bool ) ).

% Sup_empty
tff(fact_7029_Sup__empty,axiom,
    comple7399068483239264473et_nat(bot_bot_set_set_nat) = bot_bot_set_nat ).

% Sup_empty
tff(fact_7030_ccSup__empty,axiom,
    comple3096694443085538997t_real(bot_bot_set_set_real) = bot_bot_set_real ).

% ccSup_empty
tff(fact_7031_ccSup__empty,axiom,
    comple9167723193526752868t_bool(bot_bot_set_set_bool) = bot_bot_set_bool ).

% ccSup_empty
tff(fact_7032_ccSup__empty,axiom,
    comple3221217463730067765et_int(bot_bot_set_set_int) = bot_bot_set_int ).

% ccSup_empty
tff(fact_7033_ccSup__empty,axiom,
    ( pp(aa_set_bool_bool(comple6445916836919887748p_bool,bot_bot_set_bool))
  <=> bot_bot_bool ) ).

% ccSup_empty
tff(fact_7034_ccSup__empty,axiom,
    comple7399068483239264473et_nat(bot_bot_set_set_nat) = bot_bot_set_nat ).

% ccSup_empty
tff(fact_7035_SUP__bot__conv_I2_J,axiom,
    ! [B3: fun_nat_set_nat,A2: set_nat] :
      ( ( bot_bot_set_nat = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2)) )
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_set_nat(B3,X4) = bot_bot_set_nat ) ) ) ).

% SUP_bot_conv(2)
tff(fact_7036_SUP__bot__conv_I1_J,axiom,
    ! [B3: fun_nat_set_nat,A2: set_nat] :
      ( ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2)) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_set_nat(B3,X4) = bot_bot_set_nat ) ) ) ).

% SUP_bot_conv(1)
tff(fact_7037_SUP__bot,axiom,
    ! [A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_zq),A2)) = bot_bot_set_nat ).

% SUP_bot
tff(fact_7038_ccSUP__bot,axiom,
    ! [A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_zq),A2)) = bot_bot_set_nat ).

% ccSUP_bot
tff(fact_7039_SUP__const,axiom,
    ! [A2: set_real,F: set_nat] :
      ( ( A2 != bot_bot_set_real )
     => ( comple7399068483239264473et_nat(image_real_set_nat(aTP_Lamm_zr(F),A2)) = F ) ) ).

% SUP_const
tff(fact_7040_SUP__const,axiom,
    ! [A2: set_bool,F: set_nat] :
      ( ( A2 != bot_bot_set_bool )
     => ( comple7399068483239264473et_nat(image_bool_set_nat(aTP_Lamm_zs(F),A2)) = F ) ) ).

% SUP_const
tff(fact_7041_SUP__const,axiom,
    ! [A2: set_nat,F: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_zt(F)),A2)) = F ) ) ).

% SUP_const
tff(fact_7042_SUP__const,axiom,
    ! [A2: set_int,F: set_nat] :
      ( ( A2 != bot_bot_set_int )
     => ( comple7399068483239264473et_nat(image_int_set_nat(aTP_Lamm_zu(F),A2)) = F ) ) ).

% SUP_const
tff(fact_7043_ccSUP__const,axiom,
    ! [A2: set_real,F: set_nat] :
      ( ( A2 != bot_bot_set_real )
     => ( comple7399068483239264473et_nat(image_real_set_nat(aTP_Lamm_zr(F),A2)) = F ) ) ).

% ccSUP_const
tff(fact_7044_ccSUP__const,axiom,
    ! [A2: set_bool,F: set_nat] :
      ( ( A2 != bot_bot_set_bool )
     => ( comple7399068483239264473et_nat(image_bool_set_nat(aTP_Lamm_zs(F),A2)) = F ) ) ).

% ccSUP_const
tff(fact_7045_ccSUP__const,axiom,
    ! [A2: set_nat,F: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_zt(F)),A2)) = F ) ) ).

% ccSUP_const
tff(fact_7046_ccSUP__const,axiom,
    ! [A2: set_int,F: set_nat] :
      ( ( A2 != bot_bot_set_int )
     => ( comple7399068483239264473et_nat(image_int_set_nat(aTP_Lamm_zu(F),A2)) = F ) ) ).

% ccSUP_const
tff(fact_7047_UN__constant,axiom,
    ! [A2: set_real,C: set_real] :
      ( ( ( A2 = bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aTP_Lamm_zv(C),A2)) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aTP_Lamm_zv(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7048_UN__constant,axiom,
    ! [A2: set_real,C: set_bool] :
      ( ( ( A2 = bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aTP_Lamm_zw(C),A2)) = bot_bot_set_bool ) )
      & ( ( A2 != bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aTP_Lamm_zw(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7049_UN__constant,axiom,
    ! [A2: set_real,C: set_int] :
      ( ( ( A2 = bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aTP_Lamm_zx(C),A2)) = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aTP_Lamm_zx(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7050_UN__constant,axiom,
    ! [A2: set_bool,C: set_real] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aTP_Lamm_zy(C),A2)) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aTP_Lamm_zy(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7051_UN__constant,axiom,
    ! [A2: set_bool,C: set_bool] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aTP_Lamm_zz(C),A2)) = bot_bot_set_bool ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aTP_Lamm_zz(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7052_UN__constant,axiom,
    ! [A2: set_bool,C: set_int] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aTP_Lamm_aaa(C),A2)) = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aTP_Lamm_aaa(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7053_UN__constant,axiom,
    ! [A2: set_nat,C: set_real] :
      ( ( ( A2 = bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aTP_Lamm_aab(C),A2)) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aTP_Lamm_aab(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7054_UN__constant,axiom,
    ! [A2: set_nat,C: set_bool] :
      ( ( ( A2 = bot_bot_set_nat )
       => ( comple9167723193526752868t_bool(image_nat_set_bool(aTP_Lamm_aac(C),A2)) = bot_bot_set_bool ) )
      & ( ( A2 != bot_bot_set_nat )
       => ( comple9167723193526752868t_bool(image_nat_set_bool(aTP_Lamm_aac(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7055_UN__constant,axiom,
    ! [A2: set_nat,C: set_int] :
      ( ( ( A2 = bot_bot_set_nat )
       => ( comple3221217463730067765et_int(image_nat_set_int(aTP_Lamm_aad(C),A2)) = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_nat )
       => ( comple3221217463730067765et_int(image_nat_set_int(aTP_Lamm_aad(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7056_UN__constant,axiom,
    ! [A2: set_int,C: set_real] :
      ( ( ( A2 = bot_bot_set_int )
       => ( comple3096694443085538997t_real(image_int_set_real(aTP_Lamm_aae(C),A2)) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_int )
       => ( comple3096694443085538997t_real(image_int_set_real(aTP_Lamm_aae(C),A2)) = C ) ) ) ).

% UN_constant
tff(fact_7057_UN__Un,axiom,
    ! [M8: fun_nat_set_nat,A2: set_nat,B3: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(M8),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(M8),A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(M8),B3))) ).

% UN_Un
tff(fact_7058_Sup__set__def,axiom,
    ! [A2: set_set_VEBT_VEBT] : comple2820511241208326657T_VEBT(A2) = collect_VEBT_VEBT(aTP_Lamm_aaf(A2)) ).

% Sup_set_def
tff(fact_7059_Sup__set__def,axiom,
    ! [A2: set_set_real] : comple3096694443085538997t_real(A2) = collect_real(aTP_Lamm_aag(A2)) ).

% Sup_set_def
tff(fact_7060_Sup__set__def,axiom,
    ! [A2: set_set_complex] : comple8424636186594484919omplex(A2) = collect_complex(aTP_Lamm_aah(A2)) ).

% Sup_set_def
tff(fact_7061_Sup__set__def,axiom,
    ! [A2: set_set_list_nat] : comple8404747032580312297st_nat(A2) = collect_list_nat(aTP_Lamm_aai(A2)) ).

% Sup_set_def
tff(fact_7062_Sup__set__def,axiom,
    ! [A2: set_set_set_nat] : comple548664676211718543et_nat(A2) = collect_set_nat(aTP_Lamm_aaj(A2)) ).

% Sup_set_def
tff(fact_7063_Sup__set__def,axiom,
    ! [A2: set_set_int] : comple3221217463730067765et_int(A2) = collect_int(aTP_Lamm_aak(A2)) ).

% Sup_set_def
tff(fact_7064_Sup__set__def,axiom,
    ! [A2: set_set_nat] : comple7399068483239264473et_nat(A2) = collect_nat(aTP_Lamm_aal(A2)) ).

% Sup_set_def
tff(fact_7065_SUP__Sup__eq,axiom,
    ! [S3: set_set_VEBT_VEBT,X5: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(comple6106910383807688569T_bool(image_6844577774995367797T_bool(aTP_Lamm_an,S3)),X5))
    <=> pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),comple2820511241208326657T_VEBT(S3))) ) ).

% SUP_Sup_eq
tff(fact_7066_SUP__Sup__eq,axiom,
    ! [S3: set_set_real,X5: real] :
      ( pp(aa_real_bool(comple3641506650651884337l_bool(image_5130083873758588699l_bool(aTP_Lamm_ao,S3)),X5))
    <=> pp(aa_set_real_bool(member_real(X5),comple3096694443085538997t_real(S3))) ) ).

% SUP_Sup_eq
tff(fact_7067_SUP__Sup__eq,axiom,
    ! [S3: set_set_int,X5: int] :
      ( pp(aa_int_bool(comple5949612041521808305t_bool(image_7746117299667746971t_bool(aTP_Lamm_at,S3)),X5))
    <=> pp(aa_set_int_bool(member_int(X5),comple3221217463730067765et_int(S3))) ) ).

% SUP_Sup_eq
tff(fact_7068_SUP__Sup__eq,axiom,
    ! [S3: set_set_set_nat,X5: set_nat] :
      ( pp(aa_set_nat_bool(comple3837319926977133783t_bool(image_6371389588869447655t_bool(aTP_Lamm_ar,S3)),X5))
    <=> pp(aa_set_set_nat_bool(member_set_nat(X5),comple548664676211718543et_nat(S3))) ) ).

% SUP_Sup_eq
tff(fact_7069_SUP__Sup__eq,axiom,
    ! [S3: set_set_nat,X5: nat] :
      ( pp(aa_nat_bool(comple9073903007990432525t_bool(image_4478580553788107091t_bool(aTP_Lamm_as,S3)),X5))
    <=> pp(aa_set_nat_bool(member_nat(X5),comple7399068483239264473et_nat(S3))) ) ).

% SUP_Sup_eq
tff(fact_7070_SUP__UN__eq,axiom,
    ! [R2: fun_nat_set_nat,S3: set_nat,X5: nat] :
      ( pp(aa_nat_bool(comple9073903007990432525t_bool(image_6306550855257076617t_bool(aTP_Lamm_aam(R2),S3)),X5))
    <=> pp(aa_set_nat_bool(member_nat(X5),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(R2),S3)))) ) ).

% SUP_UN_eq
tff(fact_7071_SUP__Sup__eq2,axiom,
    ! [S3: set_se7855581050983116737at_nat,X5: nat,Xa2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(comple1132454847209395734t_bool(image_682314772748389261t_bool(aTP_Lamm_dp,S3)),X5),Xa2))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa2),comple5685304695842803022at_nat(S3)) ) ).

% SUP_Sup_eq2
tff(fact_7072_SUP__Sup__eq2,axiom,
    ! [S3: set_se3932177096832370463BT_nat,X5: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(comple4339735569045420674t_bool(image_7942687870447891359t_bool(aTP_Lamm_dq,S3)),X5),Xa2))
    <=> member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X5),Xa2),comple9061401370350521660BT_nat(S3)) ) ).

% SUP_Sup_eq2
tff(fact_7073_SUP__Sup__eq2,axiom,
    ! [S3: set_se6260736226359567993nt_int,X5: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(comple779371104951790942t_bool(image_8391127590443236893t_bool(aTP_Lamm_dr,S3)),X5),Xa2))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X5),Xa2),comple5382143125604098054nt_int(S3)) ) ).

% SUP_Sup_eq2
tff(fact_7074_SUP__Sup__eq2,axiom,
    ! [S3: set_se4344029326803248219nteger,X5: code_integer,Xa2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(comple5758570026194414588r_bool(image_542401931424761433r_bool(aTP_Lamm_ds,S3)),X5),Xa2))
    <=> member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X5),Xa2),comple2203973573673791208nteger(S3)) ) ).

% SUP_Sup_eq2
tff(fact_7075_SUP__Sup__eq2,axiom,
    ! [S3: set_se4826145725398303499at_num,X5: nat,Xa2: num] :
      ( pp(aa_num_bool(aa_nat_fun_num_bool(comple4326634883919016780m_bool(image_3535424960117457657m_bool(aTP_Lamm_dt,S3)),X5),Xa2))
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X5),Xa2),comple1400524285282149784at_num(S3)) ) ).

% SUP_Sup_eq2
tff(fact_7076_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      ( ( ( X6 = bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = zero_zero_nat ) )
      & ( ( X6 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X6) ) ) ) ).

% Sup_nat_def
tff(fact_7077_Inf_OINF__identity__eq,axiom,
    ! [Inf: fun_set_nat_nat,A2: set_nat] : aa_set_nat_nat(Inf,aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_dl),A2)) = aa_set_nat_nat(Inf,A2) ).

% Inf.INF_identity_eq
tff(fact_7078_Inf_OINF__identity__eq,axiom,
    ! [Inf: fun_set_int_int,A2: set_int] : aa_set_int_int(Inf,aa_set_int_set_int(image_int_int(aTP_Lamm_dm),A2)) = aa_set_int_int(Inf,A2) ).

% Inf.INF_identity_eq
tff(fact_7079_Sup_OSUP__identity__eq,axiom,
    ! [Sup: fun_set_nat_nat,A2: set_nat] : aa_set_nat_nat(Sup,aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_dl),A2)) = aa_set_nat_nat(Sup,A2) ).

% Sup.SUP_identity_eq
tff(fact_7080_Sup_OSUP__identity__eq,axiom,
    ! [Sup: fun_set_int_int,A2: set_int] : aa_set_int_int(Sup,aa_set_int_set_int(image_int_int(aTP_Lamm_dm),A2)) = aa_set_int_int(Sup,A2) ).

% Sup.SUP_identity_eq
tff(fact_7081_Sup__upper2,axiom,
    ! [U: set_int,A2: set_set_int,V: set_int] :
      ( member_set_int(U,A2)
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(V),U))
       => pp(aa_set_int_bool(ord_less_eq_set_int(V),comple3221217463730067765et_int(A2))) ) ) ).

% Sup_upper2
tff(fact_7082_Sup__upper2,axiom,
    ! [U: set_nat,A2: set_set_nat,V: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(U),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,V),U))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,V),comple7399068483239264473et_nat(A2))) ) ) ).

% Sup_upper2
tff(fact_7083_Sup__le__iff,axiom,
    ! [A2: set_set_int,B: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(A2)),B))
    <=> ! [X4: set_int] :
          ( member_set_int(X4,A2)
         => pp(aa_set_int_bool(ord_less_eq_set_int(X4),B)) ) ) ).

% Sup_le_iff
tff(fact_7084_Sup__le__iff,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(A2)),B))
    <=> ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X4),B)) ) ) ).

% Sup_le_iff
tff(fact_7085_Sup__upper,axiom,
    ! [X3: set_int,A2: set_set_int] :
      ( member_set_int(X3,A2)
     => pp(aa_set_int_bool(ord_less_eq_set_int(X3),comple3221217463730067765et_int(A2))) ) ).

% Sup_upper
tff(fact_7086_Sup__upper,axiom,
    ! [X3: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),comple7399068483239264473et_nat(A2))) ) ).

% Sup_upper
tff(fact_7087_Sup__least,axiom,
    ! [A2: set_set_int,Z: set_int] :
      ( ! [X: set_int] :
          ( member_set_int(X,A2)
         => pp(aa_set_int_bool(ord_less_eq_set_int(X),Z)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(A2)),Z)) ) ).

% Sup_least
tff(fact_7088_Sup__least,axiom,
    ! [A2: set_set_nat,Z: set_nat] :
      ( ! [X: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Z)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(A2)),Z)) ) ).

% Sup_least
tff(fact_7089_Sup__mono,axiom,
    ! [A2: set_set_int,B3: set_set_int] :
      ( ! [A3: set_int] :
          ( member_set_int(A3,A2)
         => ? [X5: set_int] :
              ( member_set_int(X5,B3)
              & pp(aa_set_int_bool(ord_less_eq_set_int(A3),X5)) ) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(A2)),comple3221217463730067765et_int(B3))) ) ).

% Sup_mono
tff(fact_7090_Sup__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ! [A3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(A3),A2))
         => ? [X5: set_nat] :
              ( pp(aa_set_set_nat_bool(member_set_nat(X5),B3))
              & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A3),X5)) ) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(A2)),comple7399068483239264473et_nat(B3))) ) ).

% Sup_mono
tff(fact_7091_Sup__eqI,axiom,
    ! [A2: set_set_int,X3: set_int] :
      ( ! [Y3: set_int] :
          ( member_set_int(Y3,A2)
         => pp(aa_set_int_bool(ord_less_eq_set_int(Y3),X3)) )
     => ( ! [Y3: set_int] :
            ( ! [Z4: set_int] :
                ( member_set_int(Z4,A2)
               => pp(aa_set_int_bool(ord_less_eq_set_int(Z4),Y3)) )
           => pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y3)) )
       => ( comple3221217463730067765et_int(A2) = X3 ) ) ) ).

% Sup_eqI
tff(fact_7092_Sup__eqI,axiom,
    ! [A2: set_set_nat,X3: set_nat] :
      ( ! [Y3: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(Y3),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Y3),X3)) )
     => ( ! [Y3: set_nat] :
            ( ! [Z4: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(Z4),A2))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Z4),Y3)) )
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Y3)) )
       => ( comple7399068483239264473et_nat(A2) = X3 ) ) ) ).

% Sup_eqI
tff(fact_7093_empty__Union__conv,axiom,
    ! [A2: set_set_real] :
      ( ( bot_bot_set_real = comple3096694443085538997t_real(A2) )
    <=> ! [X4: set_real] :
          ( member_set_real(X4,A2)
         => ( X4 = bot_bot_set_real ) ) ) ).

% empty_Union_conv
tff(fact_7094_empty__Union__conv,axiom,
    ! [A2: set_set_bool] :
      ( ( bot_bot_set_bool = comple9167723193526752868t_bool(A2) )
    <=> ! [X4: set_bool] :
          ( member_set_bool(X4,A2)
         => ( X4 = bot_bot_set_bool ) ) ) ).

% empty_Union_conv
tff(fact_7095_empty__Union__conv,axiom,
    ! [A2: set_set_int] :
      ( ( bot_bot_set_int = comple3221217463730067765et_int(A2) )
    <=> ! [X4: set_int] :
          ( member_set_int(X4,A2)
         => ( X4 = bot_bot_set_int ) ) ) ).

% empty_Union_conv
tff(fact_7096_empty__Union__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( bot_bot_set_nat = comple7399068483239264473et_nat(A2) )
    <=> ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => ( X4 = bot_bot_set_nat ) ) ) ).

% empty_Union_conv
tff(fact_7097_Union__empty__conv,axiom,
    ! [A2: set_set_real] :
      ( ( comple3096694443085538997t_real(A2) = bot_bot_set_real )
    <=> ! [X4: set_real] :
          ( member_set_real(X4,A2)
         => ( X4 = bot_bot_set_real ) ) ) ).

% Union_empty_conv
tff(fact_7098_Union__empty__conv,axiom,
    ! [A2: set_set_bool] :
      ( ( comple9167723193526752868t_bool(A2) = bot_bot_set_bool )
    <=> ! [X4: set_bool] :
          ( member_set_bool(X4,A2)
         => ( X4 = bot_bot_set_bool ) ) ) ).

% Union_empty_conv
tff(fact_7099_Union__empty__conv,axiom,
    ! [A2: set_set_int] :
      ( ( comple3221217463730067765et_int(A2) = bot_bot_set_int )
    <=> ! [X4: set_int] :
          ( member_set_int(X4,A2)
         => ( X4 = bot_bot_set_int ) ) ) ).

% Union_empty_conv
tff(fact_7100_Union__empty__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( comple7399068483239264473et_nat(A2) = bot_bot_set_nat )
    <=> ! [X4: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X4),A2))
         => ( X4 = bot_bot_set_nat ) ) ) ).

% Union_empty_conv
tff(fact_7101_Union__empty,axiom,
    comple3096694443085538997t_real(bot_bot_set_set_real) = bot_bot_set_real ).

% Union_empty
tff(fact_7102_Union__empty,axiom,
    comple9167723193526752868t_bool(bot_bot_set_set_bool) = bot_bot_set_bool ).

% Union_empty
tff(fact_7103_Union__empty,axiom,
    comple3221217463730067765et_int(bot_bot_set_set_int) = bot_bot_set_int ).

% Union_empty
tff(fact_7104_Union__empty,axiom,
    comple7399068483239264473et_nat(bot_bot_set_set_nat) = bot_bot_set_nat ).

% Union_empty
tff(fact_7105_Union__subsetI,axiom,
    ! [A2: set_set_int,B3: set_set_int] :
      ( ! [X: set_int] :
          ( member_set_int(X,A2)
         => ? [Y4: set_int] :
              ( member_set_int(Y4,B3)
              & pp(aa_set_int_bool(ord_less_eq_set_int(X),Y4)) ) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(A2)),comple3221217463730067765et_int(B3))) ) ).

% Union_subsetI
tff(fact_7106_Union__subsetI,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ! [X: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
         => ? [Y4: set_nat] :
              ( pp(aa_set_set_nat_bool(member_set_nat(Y4),B3))
              & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X),Y4)) ) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(A2)),comple7399068483239264473et_nat(B3))) ) ).

% Union_subsetI
tff(fact_7107_Union__upper,axiom,
    ! [B3: set_int,A2: set_set_int] :
      ( member_set_int(B3,A2)
     => pp(aa_set_int_bool(ord_less_eq_set_int(B3),comple3221217463730067765et_int(A2))) ) ).

% Union_upper
tff(fact_7108_Union__upper,axiom,
    ! [B3: set_nat,A2: set_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(B3),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),comple7399068483239264473et_nat(A2))) ) ).

% Union_upper
tff(fact_7109_Union__least,axiom,
    ! [A2: set_set_int,C2: set_int] :
      ( ! [X8: set_int] :
          ( member_set_int(X8,A2)
         => pp(aa_set_int_bool(ord_less_eq_set_int(X8),C2)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(A2)),C2)) ) ).

% Union_least
tff(fact_7110_Union__least,axiom,
    ! [A2: set_set_nat,C2: set_nat] :
      ( ! [X8: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X8),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X8),C2)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(A2)),C2)) ) ).

% Union_least
tff(fact_7111_Union__mono,axiom,
    ! [A2: set_set_int,B3: set_set_int] :
      ( ord_le4403425263959731960et_int(A2,B3)
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(A2)),comple3221217463730067765et_int(B3))) ) ).

% Union_mono
tff(fact_7112_Union__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(A2)),comple7399068483239264473et_nat(B3))) ) ).

% Union_mono
tff(fact_7113_SUP__commute,axiom,
    ! [F: fun_na6128946301132549280et_nat,B3: set_nat,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_aan(F),B3)),A2)) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_aap(F),A2)),B3)) ).

% SUP_commute
tff(fact_7114_UN__extend__simps_I9_J,axiom,
    ! [C2: fun_nat_set_nat,B3: fun_nat_set_nat,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aaq(C2),B3)),A2)) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(C2),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2)))) ).

% UN_extend_simps(9)
tff(fact_7115_UN__E,axiom,
    ! [B: vEBT_VEBT,B3: fun_na7453361831206303133T_VEBT,A2: set_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(B3,A2))))
     => ~ ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_nat_set_VEBT_VEBT(B3,X))) ) ) ).

% UN_E
tff(fact_7116_UN__E,axiom,
    ! [B: vEBT_VEBT,B3: fun_VE3640887651371881891T_VEBT,A2: set_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),comple2820511241208326657T_VEBT(image_2685870239581809509T_VEBT(B3,A2))))
     => ~ ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_VEB8154608084236005916T_VEBT(B3,X))) ) ) ).

% UN_E
tff(fact_7117_UN__E,axiom,
    ! [B: vEBT_VEBT,B3: fun_re4523157989745374273T_VEBT,A2: set_real] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(B3,A2))))
     => ~ ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_rea5591236203076396202T_VEBT(B3,X))) ) ) ).

% UN_E
tff(fact_7118_UN__E,axiom,
    ! [B: vEBT_VEBT,B3: fun_in7262655915974039105T_VEBT,A2: set_int] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),comple2820511241208326657T_VEBT(image_1216245965459964705T_VEBT(B3,A2))))
     => ~ ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(B),aa_int_set_VEBT_VEBT(B3,X))) ) ) ).

% UN_E
tff(fact_7119_UN__E,axiom,
    ! [B: real,B3: fun_nat_set_real,A2: set_nat] :
      ( pp(aa_set_real_bool(member_real(B),comple3096694443085538997t_real(image_nat_set_real(B3,A2))))
     => ~ ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => ~ pp(aa_set_real_bool(member_real(B),aa_nat_set_real(B3,X))) ) ) ).

% UN_E
tff(fact_7120_UN__E,axiom,
    ! [B: real,B3: fun_VE2077312251927228713t_real,A2: set_VEBT_VEBT] :
      ( pp(aa_set_real_bool(member_real(B),comple3096694443085538997t_real(image_6636839513470643793t_real(B3,A2))))
     => ~ ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => ~ pp(aa_set_real_bool(member_real(B),aa_VEB5302157898331830618t_real(B3,X))) ) ) ).

% UN_E
tff(fact_7121_UN__E,axiom,
    ! [B: real,B3: fun_real_set_real,A2: set_real] :
      ( pp(aa_set_real_bool(member_real(B),comple3096694443085538997t_real(image_real_set_real(B3,A2))))
     => ~ ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => ~ pp(aa_set_real_bool(member_real(B),aa_real_set_real(B3,X))) ) ) ).

% UN_E
tff(fact_7122_UN__E,axiom,
    ! [B: real,B3: fun_int_set_real,A2: set_int] :
      ( pp(aa_set_real_bool(member_real(B),comple3096694443085538997t_real(image_int_set_real(B3,A2))))
     => ~ ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => ~ pp(aa_set_real_bool(member_real(B),aa_int_set_real(B3,X))) ) ) ).

% UN_E
tff(fact_7123_UN__E,axiom,
    ! [B: int,B3: fun_nat_set_int,A2: set_nat] :
      ( pp(aa_set_int_bool(member_int(B),comple3221217463730067765et_int(image_nat_set_int(B3,A2))))
     => ~ ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => ~ pp(aa_set_int_bool(member_int(B),aa_nat_set_int(B3,X))) ) ) ).

% UN_E
tff(fact_7124_UN__E,axiom,
    ! [B: int,B3: fun_VE6538172513902245929et_int,A2: set_VEBT_VEBT] :
      ( pp(aa_set_int_bool(member_int(B),comple3221217463730067765et_int(image_2273570491937255121et_int(B3,A2))))
     => ~ ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => ~ pp(aa_set_int_bool(member_int(B),aa_VEBT_VEBT_set_int(B3,X))) ) ) ).

% UN_E
tff(fact_7125_UN__UN__flatten,axiom,
    ! [C2: fun_nat_set_nat,B3: fun_nat_set_nat,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(C2),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2)))) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aaq(C2),B3)),A2)) ).

% UN_UN_flatten
tff(fact_7126_SUP__eq,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_set_int,G: fun_nat_set_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,I2)),aa_nat_set_int(G,X5))) ) )
     => ( ! [J2: nat] :
            ( pp(aa_set_nat_bool(member_nat(J2),B3))
           => ? [X5: nat] :
                ( pp(aa_set_nat_bool(member_nat(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(G,J2)),aa_nat_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_nat_set_int(F,A2)) = comple3221217463730067765et_int(image_nat_set_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7127_SUP__eq,axiom,
    ! [A2: set_nat,B3: set_VEBT_VEBT,F: fun_nat_set_int,G: fun_VE6538172513902245929et_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ? [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,I2)),aa_VEBT_VEBT_set_int(G,X5))) ) )
     => ( ! [J2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(J2),B3))
           => ? [X5: nat] :
                ( pp(aa_set_nat_bool(member_nat(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(G,J2)),aa_nat_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_nat_set_int(F,A2)) = comple3221217463730067765et_int(image_2273570491937255121et_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7128_SUP__eq,axiom,
    ! [A2: set_nat,B3: set_real,F: fun_nat_set_int,G: fun_real_set_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ? [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,I2)),aa_real_set_int(G,X5))) ) )
     => ( ! [J2: real] :
            ( pp(aa_set_real_bool(member_real(J2),B3))
           => ? [X5: nat] :
                ( pp(aa_set_nat_bool(member_nat(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(G,J2)),aa_nat_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_nat_set_int(F,A2)) = comple3221217463730067765et_int(image_real_set_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7129_SUP__eq,axiom,
    ! [A2: set_nat,B3: set_int,F: fun_nat_set_int,G: fun_int_set_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,I2)),aa_int_set_int(G,X5))) ) )
     => ( ! [J2: int] :
            ( pp(aa_set_int_bool(member_int(J2),B3))
           => ? [X5: nat] :
                ( pp(aa_set_nat_bool(member_nat(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(G,J2)),aa_nat_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_nat_set_int(F,A2)) = comple3221217463730067765et_int(image_int_set_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7130_SUP__eq,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_nat,F: fun_VE6538172513902245929et_int,G: fun_nat_set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A2))
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,I2)),aa_nat_set_int(G,X5))) ) )
     => ( ! [J2: nat] :
            ( pp(aa_set_nat_bool(member_nat(J2),B3))
           => ? [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(G,J2)),aa_VEBT_VEBT_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2)) = comple3221217463730067765et_int(image_nat_set_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7131_SUP__eq,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,F: fun_VE6538172513902245929et_int,G: fun_VE6538172513902245929et_int] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A2))
         => ? [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,I2)),aa_VEBT_VEBT_set_int(G,X5))) ) )
     => ( ! [J2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(J2),B3))
           => ? [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(G,J2)),aa_VEBT_VEBT_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2)) = comple3221217463730067765et_int(image_2273570491937255121et_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7132_SUP__eq,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_real,F: fun_VE6538172513902245929et_int,G: fun_real_set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A2))
         => ? [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,I2)),aa_real_set_int(G,X5))) ) )
     => ( ! [J2: real] :
            ( pp(aa_set_real_bool(member_real(J2),B3))
           => ? [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(G,J2)),aa_VEBT_VEBT_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2)) = comple3221217463730067765et_int(image_real_set_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7133_SUP__eq,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_int,F: fun_VE6538172513902245929et_int,G: fun_int_set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A2))
         => ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,I2)),aa_int_set_int(G,X5))) ) )
     => ( ! [J2: int] :
            ( pp(aa_set_int_bool(member_int(J2),B3))
           => ? [X5: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(G,J2)),aa_VEBT_VEBT_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2)) = comple3221217463730067765et_int(image_int_set_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7134_SUP__eq,axiom,
    ! [A2: set_real,B3: set_nat,F: fun_real_set_int,G: fun_nat_set_int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(F,I2)),aa_nat_set_int(G,X5))) ) )
     => ( ! [J2: nat] :
            ( pp(aa_set_nat_bool(member_nat(J2),B3))
           => ? [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(G,J2)),aa_real_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_real_set_int(F,A2)) = comple3221217463730067765et_int(image_nat_set_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7135_SUP__eq,axiom,
    ! [A2: set_real,B3: set_VEBT_VEBT,F: fun_real_set_int,G: fun_VE6538172513902245929et_int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => ? [X5: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),B3))
              & pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(F,I2)),aa_VEBT_VEBT_set_int(G,X5))) ) )
     => ( ! [J2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(J2),B3))
           => ? [X5: real] :
                ( pp(aa_set_real_bool(member_real(X5),A2))
                & pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(G,J2)),aa_real_set_int(F,X5))) ) )
       => ( comple3221217463730067765et_int(image_real_set_int(F,A2)) = comple3221217463730067765et_int(image_2273570491937255121et_int(G,B3)) ) ) ) ).

% SUP_eq
tff(fact_7136_less__eq__Sup,axiom,
    ! [A2: set_bool,U: bool] :
      ( ! [V2: bool] :
          ( pp(aa_set_bool_bool(member_bool(V2),A2))
         => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,U),V2)) )
     => ( ( A2 != bot_bot_set_bool )
       => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,U),aa_set_bool_bool(comple6445916836919887748p_bool,A2))) ) ) ).

% less_eq_Sup
tff(fact_7137_less__eq__Sup,axiom,
    ! [A2: set_set_int,U: set_int] :
      ( ! [V2: set_int] :
          ( member_set_int(V2,A2)
         => pp(aa_set_int_bool(ord_less_eq_set_int(U),V2)) )
     => ( ( A2 != bot_bot_set_set_int )
       => pp(aa_set_int_bool(ord_less_eq_set_int(U),comple3221217463730067765et_int(A2))) ) ) ).

% less_eq_Sup
tff(fact_7138_less__eq__Sup,axiom,
    ! [A2: set_set_nat,U: set_nat] :
      ( ! [V2: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(V2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),V2)) )
     => ( ( A2 != bot_bot_set_set_nat )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),comple7399068483239264473et_nat(A2))) ) ) ).

% less_eq_Sup
tff(fact_7139_Sup__subset__mono,axiom,
    ! [A2: set_set_int,B3: set_set_int] :
      ( ord_le4403425263959731960et_int(A2,B3)
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(A2)),comple3221217463730067765et_int(B3))) ) ).

% Sup_subset_mono
tff(fact_7140_Sup__subset__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(A2)),comple7399068483239264473et_nat(B3))) ) ).

% Sup_subset_mono
tff(fact_7141_SUP__eq__const,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VE9028255658833972685et_nat,X3: set_nat] :
      ( ( I5 != bot_bo8194388402131092736T_VEBT )
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I5))
           => ( aa_VEBT_VEBT_set_nat(F,I2) = X3 ) )
       => ( comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,I5)) = X3 ) ) ) ).

% SUP_eq_const
tff(fact_7142_SUP__eq__const,axiom,
    ! [I5: set_set_nat,F: fun_set_nat_set_nat,X3: set_nat] :
      ( ( I5 != bot_bot_set_set_nat )
     => ( ! [I2: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(I2),I5))
           => ( aa_set_nat_set_nat(F,I2) = X3 ) )
       => ( comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,I5)) = X3 ) ) ) ).

% SUP_eq_const
tff(fact_7143_SUP__eq__const,axiom,
    ! [I5: set_real,F: fun_real_set_nat,X3: set_nat] :
      ( ( I5 != bot_bot_set_real )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I5))
           => ( aa_real_set_nat(F,I2) = X3 ) )
       => ( comple7399068483239264473et_nat(image_real_set_nat(F,I5)) = X3 ) ) ) ).

% SUP_eq_const
tff(fact_7144_SUP__eq__const,axiom,
    ! [I5: set_bool,F: fun_bool_set_nat,X3: set_nat] :
      ( ( I5 != bot_bot_set_bool )
     => ( ! [I2: bool] :
            ( pp(aa_set_bool_bool(member_bool(I2),I5))
           => ( aa_bool_set_nat(F,I2) = X3 ) )
       => ( comple7399068483239264473et_nat(image_bool_set_nat(F,I5)) = X3 ) ) ) ).

% SUP_eq_const
tff(fact_7145_SUP__eq__const,axiom,
    ! [I5: set_nat,F: fun_nat_set_nat,X3: set_nat] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I5))
           => ( aa_nat_set_nat(F,I2) = X3 ) )
       => ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),I5)) = X3 ) ) ) ).

% SUP_eq_const
tff(fact_7146_SUP__eq__const,axiom,
    ! [I5: set_int,F: fun_int_set_nat,X3: set_nat] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => ( aa_int_set_nat(F,I2) = X3 ) )
       => ( comple7399068483239264473et_nat(image_int_set_nat(F,I5)) = X3 ) ) ) ).

% SUP_eq_const
tff(fact_7147_SUP__eqI,axiom,
    ! [A2: set_nat,F: fun_nat_set_int,X3: set_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,I2)),X3)) )
     => ( ! [Y3: set_int] :
            ( ! [I4: nat] :
                ( pp(aa_set_nat_bool(member_nat(I4),A2))
               => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,I4)),Y3)) )
           => pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y3)) )
       => ( comple3221217463730067765et_int(image_nat_set_int(F,A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7148_SUP__eqI,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE6538172513902245929et_int,X3: set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,I2)),X3)) )
     => ( ! [Y3: set_int] :
            ( ! [I4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I4),A2))
               => pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,I4)),Y3)) )
           => pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y3)) )
       => ( comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7149_SUP__eqI,axiom,
    ! [A2: set_real,F: fun_real_set_int,X3: set_int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(F,I2)),X3)) )
     => ( ! [Y3: set_int] :
            ( ! [I4: real] :
                ( pp(aa_set_real_bool(member_real(I4),A2))
               => pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(F,I4)),Y3)) )
           => pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y3)) )
       => ( comple3221217463730067765et_int(image_real_set_int(F,A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7150_SUP__eqI,axiom,
    ! [A2: set_int,F: fun_int_set_int,X3: set_int] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(F,I2)),X3)) )
     => ( ! [Y3: set_int] :
            ( ! [I4: int] :
                ( pp(aa_set_int_bool(member_int(I4),A2))
               => pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(F,I4)),Y3)) )
           => pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y3)) )
       => ( comple3221217463730067765et_int(image_int_set_int(F,A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7151_SUP__eqI,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_set_int,X3: set_int] :
      ( ! [I2: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(F,I2)),X3)) )
     => ( ! [Y3: set_int] :
            ( ! [I4: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(I4),A2))
               => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(F,I4)),Y3)) )
           => pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y3)) )
       => ( comple3221217463730067765et_int(image_3739036796817536367et_int(F,A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7152_SUP__eqI,axiom,
    ! [A2: set_nat,F: fun_nat_set_nat,X3: set_nat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,I2)),X3)) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: nat] :
                ( pp(aa_set_nat_bool(member_nat(I4),A2))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,I4)),Y3)) )
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Y3)) )
       => ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7153_SUP__eqI,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE9028255658833972685et_nat,X3: set_nat] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(F,I2)),X3)) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I4),A2))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(F,I4)),Y3)) )
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Y3)) )
       => ( comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7154_SUP__eqI,axiom,
    ! [A2: set_real,F: fun_real_set_nat,X3: set_nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,I2)),X3)) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: real] :
                ( pp(aa_set_real_bool(member_real(I4),A2))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,I4)),Y3)) )
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Y3)) )
       => ( comple7399068483239264473et_nat(image_real_set_nat(F,A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7155_SUP__eqI,axiom,
    ! [A2: set_int,F: fun_int_set_nat,X3: set_nat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_int_set_nat(F,I2)),X3)) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: int] :
                ( pp(aa_set_int_bool(member_int(I4),A2))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_int_set_nat(F,I4)),Y3)) )
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Y3)) )
       => ( comple7399068483239264473et_nat(image_int_set_nat(F,A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7156_SUP__eqI,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_set_nat,X3: set_nat] :
      ( ! [I2: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(F,I2)),X3)) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: set_nat] :
                ( pp(aa_set_set_nat_bool(member_set_nat(I4),A2))
               => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(F,I4)),Y3)) )
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X3),Y3)) )
       => ( comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,A2)) = X3 ) ) ) ).

% SUP_eqI
tff(fact_7157_SUP__mono,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( ! [N2: nat] :
          ( pp(aa_set_nat_bool(member_nat(N2),A2))
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B3))
              & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,N2)),aa_nat_set_nat(G,X5))) ) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B3)))) ) ).

% SUP_mono
tff(fact_7158_SUP__mono,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_nat,F: fun_VE9028255658833972685et_nat,G: fun_nat_set_nat] :
      ( ! [N2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(N2),A2))
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B3))
              & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(F,N2)),aa_nat_set_nat(G,X5))) ) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B3)))) ) ).

% SUP_mono
tff(fact_7159_SUP__mono,axiom,
    ! [A2: set_real,B3: set_nat,F: fun_real_set_nat,G: fun_nat_set_nat] :
      ( ! [N2: real] :
          ( pp(aa_set_real_bool(member_real(N2),A2))
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B3))
              & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,N2)),aa_nat_set_nat(G,X5))) ) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_real_set_nat(F,A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B3)))) ) ).

% SUP_mono
tff(fact_7160_SUP__mono,axiom,
    ! [A2: set_int,B3: set_nat,F: fun_int_set_nat,G: fun_nat_set_nat] :
      ( ! [N2: int] :
          ( pp(aa_set_int_bool(member_int(N2),A2))
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B3))
              & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_int_set_nat(F,N2)),aa_nat_set_nat(G,X5))) ) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_int_set_nat(F,A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B3)))) ) ).

% SUP_mono
tff(fact_7161_SUP__mono,axiom,
    ! [A2: set_set_nat,B3: set_nat,F: fun_set_nat_set_nat,G: fun_nat_set_nat] :
      ( ! [N2: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(N2),A2))
         => ? [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),B3))
              & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(F,N2)),aa_nat_set_nat(G,X5))) ) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B3)))) ) ).

% SUP_mono
tff(fact_7162_SUP__least,axiom,
    ! [A2: set_nat,F: fun_nat_set_int,U: set_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,I2)),U)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_nat_set_int(F,A2))),U)) ) ).

% SUP_least
tff(fact_7163_SUP__least,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE6538172513902245929et_int,U: set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,I2)),U)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2))),U)) ) ).

% SUP_least
tff(fact_7164_SUP__least,axiom,
    ! [A2: set_real,F: fun_real_set_int,U: set_int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(F,I2)),U)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_real_set_int(F,A2))),U)) ) ).

% SUP_least
tff(fact_7165_SUP__least,axiom,
    ! [A2: set_int,F: fun_int_set_int,U: set_int] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(F,I2)),U)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_int_set_int(F,A2))),U)) ) ).

% SUP_least
tff(fact_7166_SUP__least,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_set_int,U: set_int] :
      ( ! [I2: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(I2),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(F,I2)),U)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_3739036796817536367et_int(F,A2))),U)) ) ).

% SUP_least
tff(fact_7167_SUP__least,axiom,
    ! [A2: set_nat,F: fun_nat_set_nat,U: set_nat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,I2)),U)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))),U)) ) ).

% SUP_least
tff(fact_7168_SUP__least,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE9028255658833972685et_nat,U: set_nat] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(F,I2)),U)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2))),U)) ) ).

% SUP_least
tff(fact_7169_SUP__least,axiom,
    ! [A2: set_real,F: fun_real_set_nat,U: set_nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,I2)),U)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_real_set_nat(F,A2))),U)) ) ).

% SUP_least
tff(fact_7170_SUP__least,axiom,
    ! [A2: set_int,F: fun_int_set_nat,U: set_nat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_int_set_nat(F,I2)),U)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_int_set_nat(F,A2))),U)) ) ).

% SUP_least
tff(fact_7171_SUP__least,axiom,
    ! [A2: set_set_nat,F: fun_set_nat_set_nat,U: set_nat] :
      ( ! [I2: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(I2),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(F,I2)),U)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,A2))),U)) ) ).

% SUP_least
tff(fact_7172_SUP__mono_H,axiom,
    ! [F: fun_nat_set_nat,G: fun_nat_set_nat,A2: set_nat] :
      ( ! [X: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,X)),aa_nat_set_nat(G,X)))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),A2)))) ) ).

% SUP_mono'
tff(fact_7173_SUP__upper,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_set_int] :
      ( pp(aa_set_nat_bool(member_nat(I),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,I)),comple3221217463730067765et_int(image_nat_set_int(F,A2)))) ) ).

% SUP_upper
tff(fact_7174_SUP__upper,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VE6538172513902245929et_int] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,I)),comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2)))) ) ).

% SUP_upper
tff(fact_7175_SUP__upper,axiom,
    ! [I: real,A2: set_real,F: fun_real_set_int] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(F,I)),comple3221217463730067765et_int(image_real_set_int(F,A2)))) ) ).

% SUP_upper
tff(fact_7176_SUP__upper,axiom,
    ! [I: int,A2: set_int,F: fun_int_set_int] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(F,I)),comple3221217463730067765et_int(image_int_set_int(F,A2)))) ) ).

% SUP_upper
tff(fact_7177_SUP__upper,axiom,
    ! [I: set_nat,A2: set_set_nat,F: fun_set_nat_set_int] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(F,I)),comple3221217463730067765et_int(image_3739036796817536367et_int(F,A2)))) ) ).

% SUP_upper
tff(fact_7178_SUP__upper,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_set_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,I)),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)))) ) ).

% SUP_upper
tff(fact_7179_SUP__upper,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VE9028255658833972685et_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(F,I)),comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2)))) ) ).

% SUP_upper
tff(fact_7180_SUP__upper,axiom,
    ! [I: real,A2: set_real,F: fun_real_set_nat] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,I)),comple7399068483239264473et_nat(image_real_set_nat(F,A2)))) ) ).

% SUP_upper
tff(fact_7181_SUP__upper,axiom,
    ! [I: int,A2: set_int,F: fun_int_set_nat] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_int_set_nat(F,I)),comple7399068483239264473et_nat(image_int_set_nat(F,A2)))) ) ).

% SUP_upper
tff(fact_7182_SUP__upper,axiom,
    ! [I: set_nat,A2: set_set_nat,F: fun_set_nat_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(F,I)),comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,A2)))) ) ).

% SUP_upper
tff(fact_7183_SUP__le__iff,axiom,
    ! [F: fun_nat_set_nat,A2: set_nat,U: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))),U))
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,X4)),U)) ) ) ).

% SUP_le_iff
tff(fact_7184_SUP__upper2,axiom,
    ! [I: nat,A2: set_nat,U: set_int,F: fun_nat_set_int] :
      ( pp(aa_set_nat_bool(member_nat(I),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(U),aa_nat_set_int(F,I)))
       => pp(aa_set_int_bool(ord_less_eq_set_int(U),comple3221217463730067765et_int(image_nat_set_int(F,A2)))) ) ) ).

% SUP_upper2
tff(fact_7185_SUP__upper2,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,U: set_int,F: fun_VE6538172513902245929et_int] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(U),aa_VEBT_VEBT_set_int(F,I)))
       => pp(aa_set_int_bool(ord_less_eq_set_int(U),comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2)))) ) ) ).

% SUP_upper2
tff(fact_7186_SUP__upper2,axiom,
    ! [I: real,A2: set_real,U: set_int,F: fun_real_set_int] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(U),aa_real_set_int(F,I)))
       => pp(aa_set_int_bool(ord_less_eq_set_int(U),comple3221217463730067765et_int(image_real_set_int(F,A2)))) ) ) ).

% SUP_upper2
tff(fact_7187_SUP__upper2,axiom,
    ! [I: int,A2: set_int,U: set_int,F: fun_int_set_int] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(U),aa_int_set_int(F,I)))
       => pp(aa_set_int_bool(ord_less_eq_set_int(U),comple3221217463730067765et_int(image_int_set_int(F,A2)))) ) ) ).

% SUP_upper2
tff(fact_7188_SUP__upper2,axiom,
    ! [I: set_nat,A2: set_set_nat,U: set_int,F: fun_set_nat_set_int] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),A2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(U),aa_set_nat_set_int(F,I)))
       => pp(aa_set_int_bool(ord_less_eq_set_int(U),comple3221217463730067765et_int(image_3739036796817536367et_int(F,A2)))) ) ) ).

% SUP_upper2
tff(fact_7189_SUP__upper2,axiom,
    ! [I: nat,A2: set_nat,U: set_nat,F: fun_nat_set_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),aa_nat_set_nat(F,I)))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)))) ) ) ).

% SUP_upper2
tff(fact_7190_SUP__upper2,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,U: set_nat,F: fun_VE9028255658833972685et_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),aa_VEBT_VEBT_set_nat(F,I)))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2)))) ) ) ).

% SUP_upper2
tff(fact_7191_SUP__upper2,axiom,
    ! [I: real,A2: set_real,U: set_nat,F: fun_real_set_nat] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),aa_real_set_nat(F,I)))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),comple7399068483239264473et_nat(image_real_set_nat(F,A2)))) ) ) ).

% SUP_upper2
tff(fact_7192_SUP__upper2,axiom,
    ! [I: int,A2: set_int,U: set_nat,F: fun_int_set_nat] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),aa_int_set_nat(F,I)))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),comple7399068483239264473et_nat(image_int_set_nat(F,A2)))) ) ) ).

% SUP_upper2
tff(fact_7193_SUP__upper2,axiom,
    ! [I: set_nat,A2: set_set_nat,U: set_nat,F: fun_set_nat_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(I),A2))
     => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),aa_set_nat_set_nat(F,I)))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,U),comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,A2)))) ) ) ).

% SUP_upper2
tff(fact_7194_SUP__lessD,axiom,
    ! [F: fun_nat_set_nat,A2: set_nat,Y: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))),Y))
     => ( pp(aa_set_nat_bool(member_nat(I),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_nat_set_nat(F,I)),Y)) ) ) ).

% SUP_lessD
tff(fact_7195_SUP__lessD,axiom,
    ! [F: fun_VE9028255658833972685et_nat,A2: set_VEBT_VEBT,Y: set_nat,I: vEBT_VEBT] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2))),Y))
     => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_VEBT_VEBT_set_nat(F,I)),Y)) ) ) ).

% SUP_lessD
tff(fact_7196_SUP__lessD,axiom,
    ! [F: fun_real_set_nat,A2: set_real,Y: set_nat,I: real] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,comple7399068483239264473et_nat(image_real_set_nat(F,A2))),Y))
     => ( pp(aa_set_real_bool(member_real(I),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_real_set_nat(F,I)),Y)) ) ) ).

% SUP_lessD
tff(fact_7197_SUP__lessD,axiom,
    ! [F: fun_int_set_nat,A2: set_int,Y: set_nat,I: int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,comple7399068483239264473et_nat(image_int_set_nat(F,A2))),Y))
     => ( pp(aa_set_int_bool(member_int(I),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_int_set_nat(F,I)),Y)) ) ) ).

% SUP_lessD
tff(fact_7198_SUP__lessD,axiom,
    ! [F: fun_set_nat_set_nat,A2: set_set_nat,Y: set_nat,I: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,A2))),Y))
     => ( pp(aa_set_set_nat_bool(member_set_nat(I),A2))
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,aa_set_nat_set_nat(F,I)),Y)) ) ) ).

% SUP_lessD
tff(fact_7199_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [F: fun_nat_set_nat,A2: set_nat,G: fun_nat_set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),A2))) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aar(F),G)),A2)) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_7200_SUP__absorb,axiom,
    ! [K: nat,I5: set_nat,A2: fun_nat_set_nat] :
      ( pp(aa_set_nat_bool(member_nat(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(A2,K)),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),I5))) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),I5)) ) ) ).

% SUP_absorb
tff(fact_7201_SUP__absorb,axiom,
    ! [K: vEBT_VEBT,I5: set_VEBT_VEBT,A2: fun_VE9028255658833972685et_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_VEBT_VEBT_set_nat(A2,K)),comple7399068483239264473et_nat(image_6451421511446451829et_nat(A2,I5))) = comple7399068483239264473et_nat(image_6451421511446451829et_nat(A2,I5)) ) ) ).

% SUP_absorb
tff(fact_7202_SUP__absorb,axiom,
    ! [K: real,I5: set_real,A2: fun_real_set_nat] :
      ( pp(aa_set_real_bool(member_real(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_real_set_nat(A2,K)),comple7399068483239264473et_nat(image_real_set_nat(A2,I5))) = comple7399068483239264473et_nat(image_real_set_nat(A2,I5)) ) ) ).

% SUP_absorb
tff(fact_7203_SUP__absorb,axiom,
    ! [K: int,I5: set_int,A2: fun_int_set_nat] :
      ( pp(aa_set_int_bool(member_int(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_int_set_nat(A2,K)),comple7399068483239264473et_nat(image_int_set_nat(A2,I5))) = comple7399068483239264473et_nat(image_int_set_nat(A2,I5)) ) ) ).

% SUP_absorb
tff(fact_7204_SUP__absorb,axiom,
    ! [K: set_nat,I5: set_set_nat,A2: fun_set_nat_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(A2,K)),comple7399068483239264473et_nat(image_7916887816326733075et_nat(A2,I5))) = comple7399068483239264473et_nat(image_7916887816326733075et_nat(A2,I5)) ) ) ).

% SUP_absorb
tff(fact_7205_UN__extend__simps_I10_J,axiom,
    ! [B3: fun_int_set_nat,F: fun_int_int,A2: set_int] : comple7399068483239264473et_nat(image_int_set_nat(aa_fun2608467900932457282et_nat(aTP_Lamm_aas(B3),F),A2)) = comple7399068483239264473et_nat(image_int_set_nat(B3,aa_set_int_set_int(image_int_int(F),A2))) ).

% UN_extend_simps(10)
tff(fact_7206_UN__extend__simps_I10_J,axiom,
    ! [B3: fun_real_set_nat,F: fun_nat_real,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun7471991306178591626et_nat(aTP_Lamm_aat(B3),F)),A2)) = comple7399068483239264473et_nat(image_real_set_nat(B3,aa_set_nat_set_real(image_nat_real(F),A2))) ).

% UN_extend_simps(10)
tff(fact_7207_UN__extend__simps_I10_J,axiom,
    ! [B3: fun_set_nat_set_nat,F: fun_nat_set_nat,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aau(B3),F)),A2)) = comple7399068483239264473et_nat(image_7916887816326733075et_nat(B3,aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))) ).

% UN_extend_simps(10)
tff(fact_7208_UN__extend__simps_I10_J,axiom,
    ! [B3: fun_int_set_nat,F: fun_nat_int,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun3079086916031139466et_nat(aTP_Lamm_aav(B3),F)),A2)) = comple7399068483239264473et_nat(image_int_set_nat(B3,aa_set_nat_set_int(image_nat_int(F),A2))) ).

% UN_extend_simps(10)
tff(fact_7209_UN__extend__simps_I10_J,axiom,
    ! [B3: fun_nat_set_nat,F: fun_nat_nat,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun7909995953855020590et_nat(aTP_Lamm_aaw(B3),F)),A2)) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),aa_set_nat_set_nat(image_nat_nat(F),A2))) ).

% UN_extend_simps(10)
tff(fact_7210_image__UN,axiom,
    ! [F: fun_nat_real,B3: fun_nat_set_nat,A2: set_nat] : aa_set_nat_set_real(image_nat_real(F),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2))) = comple3096694443085538997t_real(image_nat_set_real(aa_fun6014602394608682688t_real(aTP_Lamm_aax(F),B3),A2)) ).

% image_UN
tff(fact_7211_image__UN,axiom,
    ! [F: fun_nat_set_nat,B3: fun_nat_set_nat,A2: set_nat] : aa_set7246812951530217472et_nat(image_nat_set_nat(F),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2))) = comple548664676211718543et_nat(image_2194112158459175443et_nat(aa_fun3336736533861547162et_nat(aTP_Lamm_aay(F),B3),A2)) ).

% image_UN
tff(fact_7212_image__UN,axiom,
    ! [F: fun_nat_int,B3: fun_nat_set_nat,A2: set_nat] : aa_set_nat_set_int(image_nat_int(F),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2))) = comple3221217463730067765et_int(image_nat_set_int(aa_fun7583645416216832320et_int(aTP_Lamm_aaz(F),B3),A2)) ).

% image_UN
tff(fact_7213_image__UN,axiom,
    ! [F: fun_nat_nat,B3: fun_nat_set_nat,A2: set_nat] : aa_set_nat_set_nat(image_nat_nat(F),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2))) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aba(F),B3)),A2)) ).

% image_UN
tff(fact_7214_UN__empty2,axiom,
    ! [A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_zq),A2)) = bot_bot_set_nat ).

% UN_empty2
tff(fact_7215_UN__empty,axiom,
    ! [B3: fun_real_set_real] : comple3096694443085538997t_real(image_real_set_real(B3,bot_bot_set_real)) = bot_bot_set_real ).

% UN_empty
tff(fact_7216_UN__empty,axiom,
    ! [B3: fun_real_set_bool] : comple9167723193526752868t_bool(image_real_set_bool(B3,bot_bot_set_real)) = bot_bot_set_bool ).

% UN_empty
tff(fact_7217_UN__empty,axiom,
    ! [B3: fun_real_set_int] : comple3221217463730067765et_int(image_real_set_int(B3,bot_bot_set_real)) = bot_bot_set_int ).

% UN_empty
tff(fact_7218_UN__empty,axiom,
    ! [B3: fun_bool_set_real] : comple3096694443085538997t_real(image_bool_set_real(B3,bot_bot_set_bool)) = bot_bot_set_real ).

% UN_empty
tff(fact_7219_UN__empty,axiom,
    ! [B3: fun_bool_set_bool] : comple9167723193526752868t_bool(image_bool_set_bool(B3,bot_bot_set_bool)) = bot_bot_set_bool ).

% UN_empty
tff(fact_7220_UN__empty,axiom,
    ! [B3: fun_bool_set_int] : comple3221217463730067765et_int(image_bool_set_int(B3,bot_bot_set_bool)) = bot_bot_set_int ).

% UN_empty
tff(fact_7221_UN__empty,axiom,
    ! [B3: fun_nat_set_real] : comple3096694443085538997t_real(image_nat_set_real(B3,bot_bot_set_nat)) = bot_bot_set_real ).

% UN_empty
tff(fact_7222_UN__empty,axiom,
    ! [B3: fun_nat_set_bool] : comple9167723193526752868t_bool(image_nat_set_bool(B3,bot_bot_set_nat)) = bot_bot_set_bool ).

% UN_empty
tff(fact_7223_UN__empty,axiom,
    ! [B3: fun_nat_set_int] : comple3221217463730067765et_int(image_nat_set_int(B3,bot_bot_set_nat)) = bot_bot_set_int ).

% UN_empty
tff(fact_7224_UN__empty,axiom,
    ! [B3: fun_int_set_real] : comple3096694443085538997t_real(image_int_set_real(B3,bot_bot_set_int)) = bot_bot_set_real ).

% UN_empty
tff(fact_7225_UNION__empty__conv_I1_J,axiom,
    ! [B3: fun_nat_set_nat,A2: set_nat] :
      ( ( bot_bot_set_nat = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2)) )
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_set_nat(B3,X4) = bot_bot_set_nat ) ) ) ).

% UNION_empty_conv(1)
tff(fact_7226_UNION__empty__conv_I2_J,axiom,
    ! [B3: fun_nat_set_nat,A2: set_nat] :
      ( ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2)) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_set_nat(B3,X4) = bot_bot_set_nat ) ) ) ).

% UNION_empty_conv(2)
tff(fact_7227_UN__mono,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_set_int,G: fun_nat_set_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,X)),aa_nat_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_nat_set_int(F,A2))),comple3221217463730067765et_int(image_nat_set_int(G,B3)))) ) ) ).

% UN_mono
tff(fact_7228_UN__mono,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,F: fun_VE6538172513902245929et_int,G: fun_VE6538172513902245929et_int] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,X)),aa_VEBT_VEBT_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2))),comple3221217463730067765et_int(image_2273570491937255121et_int(G,B3)))) ) ) ).

% UN_mono
tff(fact_7229_UN__mono,axiom,
    ! [A2: set_real,B3: set_real,F: fun_real_set_int,G: fun_real_set_int] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(F,X)),aa_real_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_real_set_int(F,A2))),comple3221217463730067765et_int(image_real_set_int(G,B3)))) ) ) ).

% UN_mono
tff(fact_7230_UN__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,F: fun_set_nat_set_int,G: fun_set_nat_set_int] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
     => ( ! [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(F,X)),aa_set_nat_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_3739036796817536367et_int(F,A2))),comple3221217463730067765et_int(image_3739036796817536367et_int(G,B3)))) ) ) ).

% UN_mono
tff(fact_7231_UN__mono,axiom,
    ! [A2: set_int,B3: set_int,F: fun_int_set_int,G: fun_int_set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(F,X)),aa_int_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_int_set_int(F,A2))),comple3221217463730067765et_int(image_int_set_int(G,B3)))) ) ) ).

% UN_mono
tff(fact_7232_UN__mono,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,X)),aa_nat_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B3)))) ) ) ).

% UN_mono
tff(fact_7233_UN__mono,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,F: fun_VE9028255658833972685et_nat,G: fun_VE9028255658833972685et_nat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(F,X)),aa_VEBT_VEBT_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2))),comple7399068483239264473et_nat(image_6451421511446451829et_nat(G,B3)))) ) ) ).

% UN_mono
tff(fact_7234_UN__mono,axiom,
    ! [A2: set_real,B3: set_real,F: fun_real_set_nat,G: fun_real_set_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,X)),aa_real_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_real_set_nat(F,A2))),comple7399068483239264473et_nat(image_real_set_nat(G,B3)))) ) ) ).

% UN_mono
tff(fact_7235_UN__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,F: fun_set_nat_set_nat,G: fun_set_nat_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
     => ( ! [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(F,X)),aa_set_nat_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,A2))),comple7399068483239264473et_nat(image_7916887816326733075et_nat(G,B3)))) ) ) ).

% UN_mono
tff(fact_7236_UN__mono,axiom,
    ! [A2: set_int,B3: set_int,F: fun_int_set_nat,G: fun_int_set_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_int_set_nat(F,X)),aa_int_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_int_set_nat(F,A2))),comple7399068483239264473et_nat(image_int_set_nat(G,B3)))) ) ) ).

% UN_mono
tff(fact_7237_UN__least,axiom,
    ! [A2: set_nat,B3: fun_nat_set_int,C2: set_int] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(B3,X)),C2)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_nat_set_int(B3,A2))),C2)) ) ).

% UN_least
tff(fact_7238_UN__least,axiom,
    ! [A2: set_VEBT_VEBT,B3: fun_VE6538172513902245929et_int,C2: set_int] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(B3,X)),C2)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_2273570491937255121et_int(B3,A2))),C2)) ) ).

% UN_least
tff(fact_7239_UN__least,axiom,
    ! [A2: set_real,B3: fun_real_set_int,C2: set_int] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(B3,X)),C2)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_real_set_int(B3,A2))),C2)) ) ).

% UN_least
tff(fact_7240_UN__least,axiom,
    ! [A2: set_int,B3: fun_int_set_int,C2: set_int] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(B3,X)),C2)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_int_set_int(B3,A2))),C2)) ) ).

% UN_least
tff(fact_7241_UN__least,axiom,
    ! [A2: set_set_nat,B3: fun_set_nat_set_int,C2: set_int] :
      ( ! [X: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
         => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(B3,X)),C2)) )
     => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_3739036796817536367et_int(B3,A2))),C2)) ) ).

% UN_least
tff(fact_7242_UN__least,axiom,
    ! [A2: set_nat,B3: fun_nat_set_nat,C2: set_nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(B3,X)),C2)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2))),C2)) ) ).

% UN_least
tff(fact_7243_UN__least,axiom,
    ! [A2: set_VEBT_VEBT,B3: fun_VE9028255658833972685et_nat,C2: set_nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(B3,X)),C2)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_6451421511446451829et_nat(B3,A2))),C2)) ) ).

% UN_least
tff(fact_7244_UN__least,axiom,
    ! [A2: set_real,B3: fun_real_set_nat,C2: set_nat] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(B3,X)),C2)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_real_set_nat(B3,A2))),C2)) ) ).

% UN_least
tff(fact_7245_UN__least,axiom,
    ! [A2: set_int,B3: fun_int_set_nat,C2: set_nat] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_int_set_nat(B3,X)),C2)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_int_set_nat(B3,A2))),C2)) ) ).

% UN_least
tff(fact_7246_UN__least,axiom,
    ! [A2: set_set_nat,B3: fun_set_nat_set_nat,C2: set_nat] :
      ( ! [X: set_nat] :
          ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(B3,X)),C2)) )
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_7916887816326733075et_nat(B3,A2))),C2)) ) ).

% UN_least
tff(fact_7247_UN__upper,axiom,
    ! [A: nat,A2: set_nat,B3: fun_nat_set_int] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(B3,A)),comple3221217463730067765et_int(image_nat_set_int(B3,A2)))) ) ).

% UN_upper
tff(fact_7248_UN__upper,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B3: fun_VE6538172513902245929et_int] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(B3,A)),comple3221217463730067765et_int(image_2273570491937255121et_int(B3,A2)))) ) ).

% UN_upper
tff(fact_7249_UN__upper,axiom,
    ! [A: real,A2: set_real,B3: fun_real_set_int] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(B3,A)),comple3221217463730067765et_int(image_real_set_int(B3,A2)))) ) ).

% UN_upper
tff(fact_7250_UN__upper,axiom,
    ! [A: int,A2: set_int,B3: fun_int_set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(B3,A)),comple3221217463730067765et_int(image_int_set_int(B3,A2)))) ) ).

% UN_upper
tff(fact_7251_UN__upper,axiom,
    ! [A: set_nat,A2: set_set_nat,B3: fun_set_nat_set_int] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(B3,A)),comple3221217463730067765et_int(image_3739036796817536367et_int(B3,A2)))) ) ).

% UN_upper
tff(fact_7252_UN__upper,axiom,
    ! [A: nat,A2: set_nat,B3: fun_nat_set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(B3,A)),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),A2)))) ) ).

% UN_upper
tff(fact_7253_UN__upper,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B3: fun_VE9028255658833972685et_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(B3,A)),comple7399068483239264473et_nat(image_6451421511446451829et_nat(B3,A2)))) ) ).

% UN_upper
tff(fact_7254_UN__upper,axiom,
    ! [A: real,A2: set_real,B3: fun_real_set_nat] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(B3,A)),comple7399068483239264473et_nat(image_real_set_nat(B3,A2)))) ) ).

% UN_upper
tff(fact_7255_UN__upper,axiom,
    ! [A: int,A2: set_int,B3: fun_int_set_nat] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_int_set_nat(B3,A)),comple7399068483239264473et_nat(image_int_set_nat(B3,A2)))) ) ).

% UN_upper
tff(fact_7256_UN__upper,axiom,
    ! [A: set_nat,A2: set_set_nat,B3: fun_set_nat_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(B3,A)),comple7399068483239264473et_nat(image_7916887816326733075et_nat(B3,A2)))) ) ).

% UN_upper
tff(fact_7257_UN__subset__iff,axiom,
    ! [A2: fun_nat_set_nat,I5: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),I5))),B3))
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),I5))
         => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(A2,X4)),B3)) ) ) ).

% UN_subset_iff
tff(fact_7258_UN__insert__distrib,axiom,
    ! [U: nat,A2: set_nat,A: vEBT_VEBT,B3: fun_na7453361831206303133T_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(U),A2))
     => ( comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(aa_fun7681330223414995964T_VEBT(aTP_Lamm_zi(A),B3),A2)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7259_UN__insert__distrib,axiom,
    ! [U: nat,A2: set_nat,A: int,B3: fun_nat_set_int] :
      ( pp(aa_set_nat_bool(member_nat(U),A2))
     => ( comple3221217463730067765et_int(image_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_abb(A),B3),A2)) = aa_set_int_set_int(insert_int(A),comple3221217463730067765et_int(image_nat_set_int(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7260_UN__insert__distrib,axiom,
    ! [U: nat,A2: set_nat,A: bool,B3: fun_nat_set_bool] :
      ( pp(aa_set_nat_bool(member_nat(U),A2))
     => ( comple9167723193526752868t_bool(image_nat_set_bool(aa_fun4930178584814763708t_bool(aTP_Lamm_abc(A),B3),A2)) = aa_set_bool_set_bool(insert_bool(A),comple9167723193526752868t_bool(image_nat_set_bool(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7261_UN__insert__distrib,axiom,
    ! [U: nat,A2: set_nat,A: real,B3: fun_nat_set_real] :
      ( pp(aa_set_nat_bool(member_nat(U),A2))
     => ( comple3096694443085538997t_real(image_nat_set_real(aa_fun7593858674118267548t_real(aTP_Lamm_zj(A),B3),A2)) = aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_nat_set_real(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7262_UN__insert__distrib,axiom,
    ! [U: vEBT_VEBT,A2: set_VEBT_VEBT,A: vEBT_VEBT,B3: fun_VE3640887651371881891T_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(U),A2))
     => ( comple2820511241208326657T_VEBT(image_2685870239581809509T_VEBT(aa_fun436159583631109252T_VEBT(aTP_Lamm_abd(A),B3),A2)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_2685870239581809509T_VEBT(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7263_UN__insert__distrib,axiom,
    ! [U: vEBT_VEBT,A2: set_VEBT_VEBT,A: int,B3: fun_VE6538172513902245929et_int] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(U),A2))
     => ( comple3221217463730067765et_int(image_2273570491937255121et_int(aa_fun7169446219367245564et_int(aTP_Lamm_abe(A),B3),A2)) = aa_set_int_set_int(insert_int(A),comple3221217463730067765et_int(image_2273570491937255121et_int(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7264_UN__insert__distrib,axiom,
    ! [U: vEBT_VEBT,A2: set_VEBT_VEBT,A: bool,B3: fun_VE4526288134577429638t_bool] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(U),A2))
     => ( comple9167723193526752868t_bool(image_778176921144289224t_bool(aa_fun2602802399509281482t_bool(aTP_Lamm_abf(A),B3),A2)) = aa_set_bool_set_bool(insert_bool(A),comple9167723193526752868t_bool(image_778176921144289224t_bool(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7265_UN__insert__distrib,axiom,
    ! [U: vEBT_VEBT,A2: set_VEBT_VEBT,A: real,B3: fun_VE2077312251927228713t_real] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(U),A2))
     => ( comple3096694443085538997t_real(image_6636839513470643793t_real(aa_fun6328685536076230396t_real(aTP_Lamm_abg(A),B3),A2)) = aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_6636839513470643793t_real(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7266_UN__insert__distrib,axiom,
    ! [U: real,A2: set_real,A: vEBT_VEBT,B3: fun_re4523157989745374273T_VEBT] :
      ( pp(aa_set_real_bool(member_real(U),A2))
     => ( comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(aa_fun7713332347920806140T_VEBT(aTP_Lamm_za(A),B3),A2)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7267_UN__insert__distrib,axiom,
    ! [U: real,A2: set_real,A: int,B3: fun_real_set_int] :
      ( pp(aa_set_real_bool(member_real(U),A2))
     => ( comple3221217463730067765et_int(image_real_set_int(aa_fun7998497282572583252et_int(aTP_Lamm_zd(A),B3),A2)) = aa_set_int_set_int(insert_int(A),comple3221217463730067765et_int(image_real_set_int(B3,A2))) ) ) ).

% UN_insert_distrib
tff(fact_7268_UN__extend__simps_I6_J,axiom,
    ! [A2: fun_nat_set_nat,C2: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),C2))),B3) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_abh(A2),B3)),C2)) ).

% UN_extend_simps(6)
tff(fact_7269_UN__absorb,axiom,
    ! [K: nat,I5: set_nat,A2: fun_nat_set_nat] :
      ( pp(aa_set_nat_bool(member_nat(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(A2,K)),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),I5))) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),I5)) ) ) ).

% UN_absorb
tff(fact_7270_UN__absorb,axiom,
    ! [K: vEBT_VEBT,I5: set_VEBT_VEBT,A2: fun_VE9028255658833972685et_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_VEBT_VEBT_set_nat(A2,K)),comple7399068483239264473et_nat(image_6451421511446451829et_nat(A2,I5))) = comple7399068483239264473et_nat(image_6451421511446451829et_nat(A2,I5)) ) ) ).

% UN_absorb
tff(fact_7271_UN__absorb,axiom,
    ! [K: real,I5: set_real,A2: fun_real_set_nat] :
      ( pp(aa_set_real_bool(member_real(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_real_set_nat(A2,K)),comple7399068483239264473et_nat(image_real_set_nat(A2,I5))) = comple7399068483239264473et_nat(image_real_set_nat(A2,I5)) ) ) ).

% UN_absorb
tff(fact_7272_UN__absorb,axiom,
    ! [K: int,I5: set_int,A2: fun_int_set_nat] :
      ( pp(aa_set_int_bool(member_int(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_int_set_nat(A2,K)),comple7399068483239264473et_nat(image_int_set_nat(A2,I5))) = comple7399068483239264473et_nat(image_int_set_nat(A2,I5)) ) ) ).

% UN_absorb
tff(fact_7273_UN__absorb,axiom,
    ! [K: set_nat,I5: set_set_nat,A2: fun_set_nat_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(K),I5))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(A2,K)),comple7399068483239264473et_nat(image_7916887816326733075et_nat(A2,I5))) = comple7399068483239264473et_nat(image_7916887816326733075et_nat(A2,I5)) ) ) ).

% UN_absorb
tff(fact_7274_UN__Un__distrib,axiom,
    ! [A2: fun_nat_set_nat,B3: fun_nat_set_nat,I5: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aar(A2),B3)),I5)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),I5))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),I5))) ).

% UN_Un_distrib
tff(fact_7275_Un__Union__image,axiom,
    ! [A2: fun_nat_set_nat,B3: fun_nat_set_nat,C2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aar(A2),B3)),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),C2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),C2))) ).

% Un_Union_image
tff(fact_7276_image__Union,axiom,
    ! [F: fun_int_int,S3: set_set_int] : aa_set_int_set_int(image_int_int(F),comple3221217463730067765et_int(S3)) = comple3221217463730067765et_int(image_524474410958335435et_int(image_int_int(F),S3)) ).

% image_Union
tff(fact_7277_image__Union,axiom,
    ! [F: fun_nat_real,S3: set_set_nat] : aa_set_nat_set_real(image_nat_real(F),comple7399068483239264473et_nat(S3)) = comple3096694443085538997t_real(image_6333053925516494319t_real(image_nat_real(F),S3)) ).

% image_Union
tff(fact_7278_image__Union,axiom,
    ! [F: fun_nat_set_nat,S3: set_set_nat] : aa_set7246812951530217472et_nat(image_nat_set_nat(F),comple7399068483239264473et_nat(S3)) = comple548664676211718543et_nat(image_6725021117256019401et_nat(image_nat_set_nat(F),S3)) ).

% image_Union
tff(fact_7279_image__Union,axiom,
    ! [F: fun_nat_int,S3: set_set_nat] : aa_set_nat_set_int(image_nat_int(F),comple7399068483239264473et_nat(S3)) = comple3221217463730067765et_int(image_3739036796817536367et_int(image_nat_int(F),S3)) ).

% image_Union
tff(fact_7280_image__Union,axiom,
    ! [F: fun_nat_nat,S3: set_set_nat] : aa_set_nat_set_nat(image_nat_nat(F),comple7399068483239264473et_nat(S3)) = comple7399068483239264473et_nat(image_7916887816326733075et_nat(image_nat_nat(F),S3)) ).

% image_Union
tff(fact_7281_UN__extend__simps_I8_J,axiom,
    ! [B3: fun_nat_set_nat,A2: set_set_nat] : comple7399068483239264473et_nat(image_7916887816326733075et_nat(aTP_Lamm_abi(B3),A2)) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),comple7399068483239264473et_nat(A2))) ).

% UN_extend_simps(8)
tff(fact_7282_SUP__eq__iff,axiom,
    ! [I5: set_VEBT_VEBT,C: set_int,F: fun_VE6538172513902245929et_int] :
      ( ( I5 != bot_bo8194388402131092736T_VEBT )
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I5))
           => pp(aa_set_int_bool(ord_less_eq_set_int(C),aa_VEBT_VEBT_set_int(F,I2))) )
       => ( ( comple3221217463730067765et_int(image_2273570491937255121et_int(F,I5)) = C )
        <=> ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),I5))
             => ( aa_VEBT_VEBT_set_int(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7283_SUP__eq__iff,axiom,
    ! [I5: set_real,C: set_int,F: fun_real_set_int] :
      ( ( I5 != bot_bot_set_real )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I5))
           => pp(aa_set_int_bool(ord_less_eq_set_int(C),aa_real_set_int(F,I2))) )
       => ( ( comple3221217463730067765et_int(image_real_set_int(F,I5)) = C )
        <=> ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),I5))
             => ( aa_real_set_int(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7284_SUP__eq__iff,axiom,
    ! [I5: set_bool,C: set_int,F: fun_bool_set_int] :
      ( ( I5 != bot_bot_set_bool )
     => ( ! [I2: bool] :
            ( pp(aa_set_bool_bool(member_bool(I2),I5))
           => pp(aa_set_int_bool(ord_less_eq_set_int(C),aa_bool_set_int(F,I2))) )
       => ( ( comple3221217463730067765et_int(image_bool_set_int(F,I5)) = C )
        <=> ! [X4: bool] :
              ( pp(aa_set_bool_bool(member_bool(X4),I5))
             => ( aa_bool_set_int(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7285_SUP__eq__iff,axiom,
    ! [I5: set_nat,C: set_int,F: fun_nat_set_int] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I5))
           => pp(aa_set_int_bool(ord_less_eq_set_int(C),aa_nat_set_int(F,I2))) )
       => ( ( comple3221217463730067765et_int(image_nat_set_int(F,I5)) = C )
        <=> ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),I5))
             => ( aa_nat_set_int(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7286_SUP__eq__iff,axiom,
    ! [I5: set_int,C: set_int,F: fun_int_set_int] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => pp(aa_set_int_bool(ord_less_eq_set_int(C),aa_int_set_int(F,I2))) )
       => ( ( comple3221217463730067765et_int(image_int_set_int(F,I5)) = C )
        <=> ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),I5))
             => ( aa_int_set_int(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7287_SUP__eq__iff,axiom,
    ! [I5: set_VEBT_VEBT,C: set_nat,F: fun_VE9028255658833972685et_nat] :
      ( ( I5 != bot_bo8194388402131092736T_VEBT )
     => ( ! [I2: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I5))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),aa_VEBT_VEBT_set_nat(F,I2))) )
       => ( ( comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,I5)) = C )
        <=> ! [X4: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),I5))
             => ( aa_VEBT_VEBT_set_nat(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7288_SUP__eq__iff,axiom,
    ! [I5: set_real,C: set_nat,F: fun_real_set_nat] :
      ( ( I5 != bot_bot_set_real )
     => ( ! [I2: real] :
            ( pp(aa_set_real_bool(member_real(I2),I5))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),aa_real_set_nat(F,I2))) )
       => ( ( comple7399068483239264473et_nat(image_real_set_nat(F,I5)) = C )
        <=> ! [X4: real] :
              ( pp(aa_set_real_bool(member_real(X4),I5))
             => ( aa_real_set_nat(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7289_SUP__eq__iff,axiom,
    ! [I5: set_bool,C: set_nat,F: fun_bool_set_nat] :
      ( ( I5 != bot_bot_set_bool )
     => ( ! [I2: bool] :
            ( pp(aa_set_bool_bool(member_bool(I2),I5))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),aa_bool_set_nat(F,I2))) )
       => ( ( comple7399068483239264473et_nat(image_bool_set_nat(F,I5)) = C )
        <=> ! [X4: bool] :
              ( pp(aa_set_bool_bool(member_bool(X4),I5))
             => ( aa_bool_set_nat(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7290_SUP__eq__iff,axiom,
    ! [I5: set_nat,C: set_nat,F: fun_nat_set_nat] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I5))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),aa_nat_set_nat(F,I2))) )
       => ( ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),I5)) = C )
        <=> ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),I5))
             => ( aa_nat_set_nat(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7291_SUP__eq__iff,axiom,
    ! [I5: set_int,C: set_nat,F: fun_int_set_nat] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,C),aa_int_set_nat(F,I2))) )
       => ( ( comple7399068483239264473et_nat(image_int_set_nat(F,I5)) = C )
        <=> ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),I5))
             => ( aa_int_set_nat(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_7292_SUP__subset__mono,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_set_int,G: fun_nat_set_int] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_nat_set_int(F,X)),aa_nat_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_nat_set_int(F,A2))),comple3221217463730067765et_int(image_nat_set_int(G,B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7293_SUP__subset__mono,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,F: fun_VE6538172513902245929et_int,G: fun_VE6538172513902245929et_int] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_VEBT_VEBT_set_int(F,X)),aa_VEBT_VEBT_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_2273570491937255121et_int(F,A2))),comple3221217463730067765et_int(image_2273570491937255121et_int(G,B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7294_SUP__subset__mono,axiom,
    ! [A2: set_real,B3: set_real,F: fun_real_set_int,G: fun_real_set_int] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_real_set_int(F,X)),aa_real_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_real_set_int(F,A2))),comple3221217463730067765et_int(image_real_set_int(G,B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7295_SUP__subset__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,F: fun_set_nat_set_int,G: fun_set_nat_set_int] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
     => ( ! [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(F,X)),aa_set_nat_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_3739036796817536367et_int(F,A2))),comple3221217463730067765et_int(image_3739036796817536367et_int(G,B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7296_SUP__subset__mono,axiom,
    ! [A2: set_int,B3: set_int,F: fun_int_set_int,G: fun_int_set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => pp(aa_set_int_bool(ord_less_eq_set_int(aa_int_set_int(F,X)),aa_int_set_int(G,X))) )
       => pp(aa_set_int_bool(ord_less_eq_set_int(comple3221217463730067765et_int(image_int_set_int(F,A2))),comple3221217463730067765et_int(image_int_set_int(G,B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7297_SUP__subset__mono,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),B3))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(F,X)),aa_nat_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7298_SUP__subset__mono,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,F: fun_VE9028255658833972685et_nat,G: fun_VE9028255658833972685et_nat] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B3))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(F,X)),aa_VEBT_VEBT_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2))),comple7399068483239264473et_nat(image_6451421511446451829et_nat(G,B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7299_SUP__subset__mono,axiom,
    ! [A2: set_real,B3: set_real,F: fun_real_set_nat,G: fun_real_set_nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B3))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_real_set_nat(F,X)),aa_real_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_real_set_nat(F,A2))),comple7399068483239264473et_nat(image_real_set_nat(G,B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7300_SUP__subset__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,F: fun_set_nat_set_nat,G: fun_set_nat_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),B3))
     => ( ! [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(F,X)),aa_set_nat_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,A2))),comple7399068483239264473et_nat(image_7916887816326733075et_nat(G,B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7301_SUP__subset__mono,axiom,
    ! [A2: set_int,B3: set_int,F: fun_int_set_nat,G: fun_int_set_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_int_set_nat(F,X)),aa_int_set_nat(G,X))) )
       => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_int_set_nat(F,A2))),comple7399068483239264473et_nat(image_int_set_nat(G,B3)))) ) ) ).

% SUP_subset_mono
tff(fact_7302_SUP__constant,axiom,
    ! [A2: set_real,C: set_real] :
      ( ( ( A2 = bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aTP_Lamm_zv(C),A2)) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_real )
       => ( comple3096694443085538997t_real(image_real_set_real(aTP_Lamm_zv(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7303_SUP__constant,axiom,
    ! [A2: set_real,C: set_bool] :
      ( ( ( A2 = bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aTP_Lamm_zw(C),A2)) = bot_bot_set_bool ) )
      & ( ( A2 != bot_bot_set_real )
       => ( comple9167723193526752868t_bool(image_real_set_bool(aTP_Lamm_zw(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7304_SUP__constant,axiom,
    ! [A2: set_real,C: set_int] :
      ( ( ( A2 = bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aTP_Lamm_zx(C),A2)) = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_real )
       => ( comple3221217463730067765et_int(image_real_set_int(aTP_Lamm_zx(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7305_SUP__constant,axiom,
    ! [A2: set_bool,C: set_real] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aTP_Lamm_zy(C),A2)) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( comple3096694443085538997t_real(image_bool_set_real(aTP_Lamm_zy(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7306_SUP__constant,axiom,
    ! [A2: set_bool,C: set_bool] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aTP_Lamm_zz(C),A2)) = bot_bot_set_bool ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( comple9167723193526752868t_bool(image_bool_set_bool(aTP_Lamm_zz(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7307_SUP__constant,axiom,
    ! [A2: set_bool,C: set_int] :
      ( ( ( A2 = bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aTP_Lamm_aaa(C),A2)) = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_bool )
       => ( comple3221217463730067765et_int(image_bool_set_int(aTP_Lamm_aaa(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7308_SUP__constant,axiom,
    ! [A2: set_nat,C: set_real] :
      ( ( ( A2 = bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aTP_Lamm_aab(C),A2)) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_nat )
       => ( comple3096694443085538997t_real(image_nat_set_real(aTP_Lamm_aab(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7309_SUP__constant,axiom,
    ! [A2: set_nat,C: set_bool] :
      ( ( ( A2 = bot_bot_set_nat )
       => ( comple9167723193526752868t_bool(image_nat_set_bool(aTP_Lamm_aac(C),A2)) = bot_bot_set_bool ) )
      & ( ( A2 != bot_bot_set_nat )
       => ( comple9167723193526752868t_bool(image_nat_set_bool(aTP_Lamm_aac(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7310_SUP__constant,axiom,
    ! [A2: set_nat,C: set_int] :
      ( ( ( A2 = bot_bot_set_nat )
       => ( comple3221217463730067765et_int(image_nat_set_int(aTP_Lamm_aad(C),A2)) = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_nat )
       => ( comple3221217463730067765et_int(image_nat_set_int(aTP_Lamm_aad(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7311_SUP__constant,axiom,
    ! [A2: set_int,C: set_real] :
      ( ( ( A2 = bot_bot_set_int )
       => ( comple3096694443085538997t_real(image_int_set_real(aTP_Lamm_aae(C),A2)) = bot_bot_set_real ) )
      & ( ( A2 != bot_bot_set_int )
       => ( comple3096694443085538997t_real(image_int_set_real(aTP_Lamm_aae(C),A2)) = C ) ) ) ).

% SUP_constant
tff(fact_7312_SUP__empty,axiom,
    ! [F: fun_real_set_real] : comple3096694443085538997t_real(image_real_set_real(F,bot_bot_set_real)) = bot_bot_set_real ).

% SUP_empty
tff(fact_7313_SUP__empty,axiom,
    ! [F: fun_real_set_bool] : comple9167723193526752868t_bool(image_real_set_bool(F,bot_bot_set_real)) = bot_bot_set_bool ).

% SUP_empty
tff(fact_7314_SUP__empty,axiom,
    ! [F: fun_real_set_int] : comple3221217463730067765et_int(image_real_set_int(F,bot_bot_set_real)) = bot_bot_set_int ).

% SUP_empty
tff(fact_7315_SUP__empty,axiom,
    ! [F: fun_bool_set_real] : comple3096694443085538997t_real(image_bool_set_real(F,bot_bot_set_bool)) = bot_bot_set_real ).

% SUP_empty
tff(fact_7316_SUP__empty,axiom,
    ! [F: fun_bool_set_bool] : comple9167723193526752868t_bool(image_bool_set_bool(F,bot_bot_set_bool)) = bot_bot_set_bool ).

% SUP_empty
tff(fact_7317_SUP__empty,axiom,
    ! [F: fun_bool_set_int] : comple3221217463730067765et_int(image_bool_set_int(F,bot_bot_set_bool)) = bot_bot_set_int ).

% SUP_empty
tff(fact_7318_SUP__empty,axiom,
    ! [F: fun_nat_set_real] : comple3096694443085538997t_real(image_nat_set_real(F,bot_bot_set_nat)) = bot_bot_set_real ).

% SUP_empty
tff(fact_7319_SUP__empty,axiom,
    ! [F: fun_nat_set_bool] : comple9167723193526752868t_bool(image_nat_set_bool(F,bot_bot_set_nat)) = bot_bot_set_bool ).

% SUP_empty
tff(fact_7320_SUP__empty,axiom,
    ! [F: fun_nat_set_int] : comple3221217463730067765et_int(image_nat_set_int(F,bot_bot_set_nat)) = bot_bot_set_int ).

% SUP_empty
tff(fact_7321_SUP__empty,axiom,
    ! [F: fun_int_set_real] : comple3096694443085538997t_real(image_int_set_real(F,bot_bot_set_int)) = bot_bot_set_real ).

% SUP_empty
tff(fact_7322_SUP__insert,axiom,
    ! [F: fun_nat_set_nat,A: nat,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_set_nat_set_nat(insert_nat(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(F,A)),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2))) ).

% SUP_insert
tff(fact_7323_SUP__insert,axiom,
    ! [F: fun_VE9028255658833972685et_nat,A: vEBT_VEBT,A2: set_VEBT_VEBT] : comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_VEBT_VEBT_set_nat(F,A)),comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2))) ).

% SUP_insert
tff(fact_7324_SUP__insert,axiom,
    ! [F: fun_int_set_nat,A: int,A2: set_int] : comple7399068483239264473et_nat(image_int_set_nat(F,aa_set_int_set_int(insert_int(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_int_set_nat(F,A)),comple7399068483239264473et_nat(image_int_set_nat(F,A2))) ).

% SUP_insert
tff(fact_7325_SUP__insert,axiom,
    ! [F: fun_bool_set_nat,A: bool,A2: set_bool] : comple7399068483239264473et_nat(image_bool_set_nat(F,aa_set_bool_set_bool(insert_bool(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_bool_set_nat(F,A)),comple7399068483239264473et_nat(image_bool_set_nat(F,A2))) ).

% SUP_insert
tff(fact_7326_SUP__insert,axiom,
    ! [F: fun_real_set_nat,A: real,A2: set_real] : comple7399068483239264473et_nat(image_real_set_nat(F,aa_set_real_set_real(insert_real(A),A2))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_real_set_nat(F,A)),comple7399068483239264473et_nat(image_real_set_nat(F,A2))) ).

% SUP_insert
tff(fact_7327_SUP__union,axiom,
    ! [M8: fun_nat_set_nat,A2: set_nat,B3: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(M8),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(M8),A2))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(M8),B3))) ).

% SUP_union
tff(fact_7328_UN__extend__simps_I1_J,axiom,
    ! [C2: set_real,A: vEBT_VEBT,B3: fun_re4523157989745374273T_VEBT] :
      ( ( ( C2 = bot_bot_set_real )
       => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(B3,C2))) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ) )
      & ( ( C2 != bot_bot_set_real )
       => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(B3,C2))) = comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(aa_fun7713332347920806140T_VEBT(aTP_Lamm_za(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7329_UN__extend__simps_I1_J,axiom,
    ! [C2: set_real,A: real,B3: fun_real_set_real] :
      ( ( ( C2 = bot_bot_set_real )
       => ( aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_real_set_real(B3,C2))) = aa_set_real_set_real(insert_real(A),bot_bot_set_real) ) )
      & ( ( C2 != bot_bot_set_real )
       => ( aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_real_set_real(B3,C2))) = comple3096694443085538997t_real(image_real_set_real(aa_fun686715674179075668t_real(aTP_Lamm_zb(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7330_UN__extend__simps_I1_J,axiom,
    ! [C2: set_real,A: bool,B3: fun_real_set_bool] :
      ( ( ( C2 = bot_bot_set_real )
       => ( aa_set_bool_set_bool(insert_bool(A),comple9167723193526752868t_bool(image_real_set_bool(B3,C2))) = aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool) ) )
      & ( ( C2 != bot_bot_set_real )
       => ( aa_set_bool_set_bool(insert_bool(A),comple9167723193526752868t_bool(image_real_set_bool(B3,C2))) = comple9167723193526752868t_bool(image_real_set_bool(aa_fun7454438436055979324t_bool(aTP_Lamm_zc(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7331_UN__extend__simps_I1_J,axiom,
    ! [C2: set_real,A: int,B3: fun_real_set_int] :
      ( ( ( C2 = bot_bot_set_real )
       => ( aa_set_int_set_int(insert_int(A),comple3221217463730067765et_int(image_real_set_int(B3,C2))) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) )
      & ( ( C2 != bot_bot_set_real )
       => ( aa_set_int_set_int(insert_int(A),comple3221217463730067765et_int(image_real_set_int(B3,C2))) = comple3221217463730067765et_int(image_real_set_int(aa_fun7998497282572583252et_int(aTP_Lamm_zd(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7332_UN__extend__simps_I1_J,axiom,
    ! [C2: set_bool,A: vEBT_VEBT,B3: fun_bo1266566265771860096T_VEBT] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_5756441063665793602T_VEBT(B3,C2))) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_5756441063665793602T_VEBT(B3,C2))) = comple2820511241208326657T_VEBT(image_5756441063665793602T_VEBT(aa_fun8832029763029799358T_VEBT(aTP_Lamm_ze(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7333_UN__extend__simps_I1_J,axiom,
    ! [C2: set_bool,A: real,B3: fun_bool_set_real] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_bool_set_real(B3,C2))) = aa_set_real_set_real(insert_real(A),bot_bot_set_real) ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_bool_set_real(B3,C2))) = comple3096694443085538997t_real(image_bool_set_real(aa_fun418618139989480252t_real(aTP_Lamm_zf(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7334_UN__extend__simps_I1_J,axiom,
    ! [C2: set_bool,A: bool,B3: fun_bool_set_bool] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( aa_set_bool_set_bool(insert_bool(A),comple9167723193526752868t_bool(image_bool_set_bool(B3,C2))) = aa_set_bool_set_bool(insert_bool(A),bot_bot_set_bool) ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( aa_set_bool_set_bool(insert_bool(A),comple9167723193526752868t_bool(image_bool_set_bool(B3,C2))) = comple9167723193526752868t_bool(image_bool_set_bool(aa_fun2947227189488945156t_bool(aTP_Lamm_zg(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7335_UN__extend__simps_I1_J,axiom,
    ! [C2: set_bool,A: int,B3: fun_bool_set_int] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( aa_set_int_set_int(insert_int(A),comple3221217463730067765et_int(image_bool_set_int(B3,C2))) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( aa_set_int_set_int(insert_int(A),comple3221217463730067765et_int(image_bool_set_int(B3,C2))) = comple3221217463730067765et_int(image_bool_set_int(aa_fun4225254755057748412et_int(aTP_Lamm_zh(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7336_UN__extend__simps_I1_J,axiom,
    ! [C2: set_nat,A: vEBT_VEBT,B3: fun_na7453361831206303133T_VEBT] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(B3,C2))) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(B3,C2))) = comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(aa_fun7681330223414995964T_VEBT(aTP_Lamm_zi(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7337_UN__extend__simps_I1_J,axiom,
    ! [C2: set_nat,A: real,B3: fun_nat_set_real] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_nat_set_real(B3,C2))) = aa_set_real_set_real(insert_real(A),bot_bot_set_real) ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( aa_set_real_set_real(insert_real(A),comple3096694443085538997t_real(image_nat_set_real(B3,C2))) = comple3096694443085538997t_real(image_nat_set_real(aa_fun7593858674118267548t_real(aTP_Lamm_zj(A),B3),C2)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_7338_SUP__UNION,axiom,
    ! [F: fun_nat_set_nat,G: fun_nat_set_nat,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),A2)))) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aaq(F),G)),A2)) ).

% SUP_UNION
tff(fact_7339_UN__extend__simps_I3_J,axiom,
    ! [C2: set_real,A2: set_nat,B3: fun_real_set_nat] :
      ( ( ( C2 = bot_bot_set_real )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(image_real_set_nat(B3,C2))) = A2 ) )
      & ( ( C2 != bot_bot_set_real )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(image_real_set_nat(B3,C2))) = comple7399068483239264473et_nat(image_real_set_nat(aa_fun8063024115349955228et_nat(aTP_Lamm_abj(A2),B3),C2)) ) ) ) ).

% UN_extend_simps(3)
tff(fact_7340_UN__extend__simps_I3_J,axiom,
    ! [C2: set_bool,A2: set_nat,B3: fun_bool_set_nat] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(image_bool_set_nat(B3,C2))) = A2 ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(image_bool_set_nat(B3,C2))) = comple7399068483239264473et_nat(image_bool_set_nat(aa_fun7812420880633473724et_nat(aTP_Lamm_abk(A2),B3),C2)) ) ) ) ).

% UN_extend_simps(3)
tff(fact_7341_UN__extend__simps_I3_J,axiom,
    ! [C2: set_nat,A2: set_nat,B3: fun_nat_set_nat] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),C2))) = A2 ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B3),C2))) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_abl(A2),B3)),C2)) ) ) ) ).

% UN_extend_simps(3)
tff(fact_7342_UN__extend__simps_I3_J,axiom,
    ! [C2: set_int,A2: set_nat,B3: fun_int_set_nat] :
      ( ( ( C2 = bot_bot_set_int )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(image_int_set_nat(B3,C2))) = A2 ) )
      & ( ( C2 != bot_bot_set_int )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(image_int_set_nat(B3,C2))) = comple7399068483239264473et_nat(image_int_set_nat(aa_fun5565777959088800668et_nat(aTP_Lamm_abm(A2),B3),C2)) ) ) ) ).

% UN_extend_simps(3)
tff(fact_7343_UN__extend__simps_I2_J,axiom,
    ! [C2: set_real,A2: fun_real_set_nat,B3: set_nat] :
      ( ( ( C2 = bot_bot_set_real )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_real_set_nat(A2,C2))),B3) = B3 ) )
      & ( ( C2 != bot_bot_set_real )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_real_set_nat(A2,C2))),B3) = comple7399068483239264473et_nat(image_real_set_nat(aa_set7314861974372065651et_nat(aTP_Lamm_abn(A2),B3),C2)) ) ) ) ).

% UN_extend_simps(2)
tff(fact_7344_UN__extend__simps_I2_J,axiom,
    ! [C2: set_bool,A2: fun_bool_set_nat,B3: set_nat] :
      ( ( ( C2 = bot_bot_set_bool )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_bool_set_nat(A2,C2))),B3) = B3 ) )
      & ( ( C2 != bot_bot_set_bool )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_bool_set_nat(A2,C2))),B3) = comple7399068483239264473et_nat(image_bool_set_nat(aa_set647555662090937580et_nat(aTP_Lamm_abo(A2),B3),C2)) ) ) ) ).

% UN_extend_simps(2)
tff(fact_7345_UN__extend__simps_I2_J,axiom,
    ! [C2: set_nat,A2: fun_nat_set_nat,B3: set_nat] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),C2))),B3) = B3 ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A2),C2))),B3) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_abp(A2),B3)),C2)) ) ) ) ).

% UN_extend_simps(2)
tff(fact_7346_UN__extend__simps_I2_J,axiom,
    ! [C2: set_int,A2: fun_int_set_nat,B3: set_nat] :
      ( ( ( C2 = bot_bot_set_int )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_int_set_nat(A2,C2))),B3) = B3 ) )
      & ( ( C2 != bot_bot_set_int )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_int_set_nat(A2,C2))),B3) = comple7399068483239264473et_nat(image_int_set_nat(aa_set933959217715569011et_nat(aTP_Lamm_abq(A2),B3),C2)) ) ) ) ).

% UN_extend_simps(2)
tff(fact_7347_UNION__singleton__eq__range,axiom,
    ! [F: fun_nat_set_nat,A2: set_nat] : comple548664676211718543et_nat(image_2194112158459175443et_nat(aTP_Lamm_abr(F),A2)) = aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2) ).

% UNION_singleton_eq_range
tff(fact_7348_UNION__singleton__eq__range,axiom,
    ! [F: fun_nat_real,A2: set_nat] : comple3096694443085538997t_real(image_nat_set_real(aTP_Lamm_abs(F),A2)) = aa_set_nat_set_real(image_nat_real(F),A2) ).

% UNION_singleton_eq_range
tff(fact_7349_UNION__singleton__eq__range,axiom,
    ! [F: fun_nat_int,A2: set_nat] : comple3221217463730067765et_int(image_nat_set_int(aTP_Lamm_abt(F),A2)) = aa_set_nat_set_int(image_nat_int(F),A2) ).

% UNION_singleton_eq_range
tff(fact_7350_UNION__singleton__eq__range,axiom,
    ! [F: fun_int_int,A2: set_int] : comple3221217463730067765et_int(image_int_set_int(aTP_Lamm_abu(F),A2)) = aa_set_int_set_int(image_int_int(F),A2) ).

% UNION_singleton_eq_range
tff(fact_7351_UNION__singleton__eq__range,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_abv(F)),A2)) = aa_set_nat_set_nat(image_nat_nat(F),A2) ).

% UNION_singleton_eq_range
tff(fact_7352_UNION__fun__upd,axiom,
    ! [A2: fun_VE2077312251927228713t_real,I: vEBT_VEBT,B3: set_real,J4: set_VEBT_VEBT] : comple3096694443085538997t_real(image_6636839513470643793t_real(fun_up5367274378234922969t_real(A2,I,B3),J4)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,comple3096694443085538997t_real(image_6636839513470643793t_real(A2,minus_5127226145743854075T_VEBT(J4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))))),if_set_real(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),J4),B3,bot_bot_set_real)) ).

% UNION_fun_upd
tff(fact_7353_UNION__fun__upd,axiom,
    ! [A2: fun_VE4526288134577429638t_bool,I: vEBT_VEBT,B3: set_bool,J4: set_VEBT_VEBT] : comple9167723193526752868t_bool(image_778176921144289224t_bool(fun_up6850350255959026752t_bool(A2,I,B3),J4)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,comple9167723193526752868t_bool(image_778176921144289224t_bool(A2,minus_5127226145743854075T_VEBT(J4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))))),if_set_bool(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),J4),B3,bot_bot_set_bool)) ).

% UNION_fun_upd
tff(fact_7354_UNION__fun__upd,axiom,
    ! [A2: fun_VE6538172513902245929et_int,I: vEBT_VEBT,B3: set_int,J4: set_VEBT_VEBT] : comple3221217463730067765et_int(image_2273570491937255121et_int(fun_up4996286513446587481et_int(A2,I,B3),J4)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,comple3221217463730067765et_int(image_2273570491937255121et_int(A2,minus_5127226145743854075T_VEBT(J4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT))))),aa_set_int_set_int(aa_set4234024024286713313et_int(if_set_int(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),J4)),B3),bot_bot_set_int)) ).

% UNION_fun_upd
tff(fact_7355_UNION__fun__upd,axiom,
    ! [A2: fun_real_set_real,I: real,B3: set_real,J4: set_real] : comple3096694443085538997t_real(image_real_set_real(fun_up567933838727694221t_real(A2,I,B3),J4)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,comple3096694443085538997t_real(image_real_set_real(A2,minus_minus_set_real(J4,aa_set_real_set_real(insert_real(I),bot_bot_set_real))))),if_set_real(aa_set_real_bool(member_real(I),J4),B3,bot_bot_set_real)) ).

% UNION_fun_upd
tff(fact_7356_UNION__fun__upd,axiom,
    ! [A2: fun_real_set_bool,I: real,B3: set_bool,J4: set_real] : comple9167723193526752868t_bool(image_real_set_bool(fun_up3965960961845116812t_bool(A2,I,B3),J4)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,comple9167723193526752868t_bool(image_real_set_bool(A2,minus_minus_set_real(J4,aa_set_real_set_real(insert_real(I),bot_bot_set_real))))),if_set_bool(aa_set_real_bool(member_real(I),J4),B3,bot_bot_set_bool)) ).

% UNION_fun_upd
tff(fact_7357_UNION__fun__upd,axiom,
    ! [A2: fun_real_set_int,I: real,B3: set_int,J4: set_real] : comple3221217463730067765et_int(image_real_set_int(fun_upd_real_set_int(A2,I,B3),J4)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,comple3221217463730067765et_int(image_real_set_int(A2,minus_minus_set_real(J4,aa_set_real_set_real(insert_real(I),bot_bot_set_real))))),aa_set_int_set_int(aa_set4234024024286713313et_int(if_set_int(aa_set_real_bool(member_real(I),J4)),B3),bot_bot_set_int)) ).

% UNION_fun_upd
tff(fact_7358_UNION__fun__upd,axiom,
    ! [A2: fun_bool_set_real,I: bool,B3: set_real,J4: set_bool] : comple3096694443085538997t_real(image_bool_set_real(fun_up5996799812878539836t_real(A2,I,B3),J4)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,comple3096694443085538997t_real(image_bool_set_real(A2,minus_minus_set_bool(J4,aa_set_bool_set_bool(insert_bool(I),bot_bot_set_bool))))),if_set_real(aa_set_bool_bool(member_bool(I),J4),B3,bot_bot_set_real)) ).

% UNION_fun_upd
tff(fact_7359_UNION__fun__upd,axiom,
    ! [A2: fun_bool_set_bool,I: bool,B3: set_bool,J4: set_bool] : comple9167723193526752868t_bool(image_bool_set_bool(fun_up1660446765068177181t_bool(A2,I,B3),J4)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,comple9167723193526752868t_bool(image_bool_set_bool(A2,minus_minus_set_bool(J4,aa_set_bool_set_bool(insert_bool(I),bot_bot_set_bool))))),if_set_bool(aa_set_bool_bool(member_bool(I),J4),B3,bot_bot_set_bool)) ).

% UNION_fun_upd
tff(fact_7360_UNION__fun__upd,axiom,
    ! [A2: fun_bool_set_int,I: bool,B3: set_int,J4: set_bool] : comple3221217463730067765et_int(image_bool_set_int(fun_upd_bool_set_int(A2,I,B3),J4)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,comple3221217463730067765et_int(image_bool_set_int(A2,minus_minus_set_bool(J4,aa_set_bool_set_bool(insert_bool(I),bot_bot_set_bool))))),aa_set_int_set_int(aa_set4234024024286713313et_int(if_set_int(aa_set_bool_bool(member_bool(I),J4)),B3),bot_bot_set_int)) ).

% UNION_fun_upd
tff(fact_7361_UNION__fun__upd,axiom,
    ! [A2: fun_int_set_real,I: int,B3: set_real,J4: set_int] : comple3096694443085538997t_real(image_int_set_real(fun_upd_int_set_real(A2,I,B3),J4)) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,comple3096694443085538997t_real(image_int_set_real(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,J4),aa_set_int_set_int(insert_int(I),bot_bot_set_int))))),if_set_real(aa_set_int_bool(member_int(I),J4),B3,bot_bot_set_real)) ).

% UNION_fun_upd
tff(fact_7362_ccpo__Sup__singleton,axiom,
    ! [X3: bool] :
      ( pp(aa_set_bool_bool(comple6445916836919887748p_bool,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)))
    <=> pp(X3) ) ).

% ccpo_Sup_singleton
tff(fact_7363_ccpo__Sup__singleton,axiom,
    ! [X3: set_nat] : comple7399068483239264473et_nat(aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat)) = X3 ).

% ccpo_Sup_singleton
tff(fact_7364_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N: nat,X14: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_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,N),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_7365_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7366_SUP__nat__binary,axiom,
    ! [A2: set_nat,B3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_zt(B3)),collect_nat(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat))))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3) ).

% SUP_nat_binary
tff(fact_7367_Union__image__empty,axiom,
    ! [A2: set_nat,F: fun_real_set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(image_real_set_nat(F,bot_bot_set_real))) = A2 ).

% Union_image_empty
tff(fact_7368_Union__image__empty,axiom,
    ! [A2: set_nat,F: fun_bool_set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(image_bool_set_nat(F,bot_bot_set_bool))) = A2 ).

% Union_image_empty
tff(fact_7369_Union__image__empty,axiom,
    ! [A2: set_nat,F: fun_nat_set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),bot_bot_set_nat))) = A2 ).

% Union_image_empty
tff(fact_7370_Union__image__empty,axiom,
    ! [A2: set_nat,F: fun_int_set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(image_int_set_nat(F,bot_bot_set_int))) = A2 ).

% Union_image_empty
tff(fact_7371_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ).

% max.absorb1
tff(fact_7372_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb1
tff(fact_7373_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb1
tff(fact_7374_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_7375_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_7376_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(A,B)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) ) ).

% max.absorb2
tff(fact_7377_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb2
tff(fact_7378_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb2
tff(fact_7379_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_7380_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_7381_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C),A)
    <=> ( ord_le2932123472753598470d_enat(B,A)
        & ord_le2932123472753598470d_enat(C,A) ) ) ).

% max.bounded_iff
tff(fact_7382_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% max.bounded_iff
tff(fact_7383_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% max.bounded_iff
tff(fact_7384_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.bounded_iff
tff(fact_7385_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% max.bounded_iff
tff(fact_7386_max__less__iff__conj,axiom,
    ! [X3: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),Y),Z)
    <=> ( ord_le72135733267957522d_enat(X3,Z)
        & ord_le72135733267957522d_enat(Y,Z) ) ) ).

% max_less_iff_conj
tff(fact_7387_max__less__iff__conj,axiom,
    ! [X3: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ord_max_real(X3),Y)),Z))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_7388_max__less__iff__conj,axiom,
    ! [X3: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X3),Y)),Z))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_7389_max__less__iff__conj,axiom,
    ! [X3: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Y)),Z))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_7390_max__less__iff__conj,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_7391_max__less__iff__conj,axiom,
    ! [X3: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_7392_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ord_le72135733267957522d_enat(A,B)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) ) ).

% max.absorb4
tff(fact_7393_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(ord_max_real(A),B) = B ) ) ).

% max.absorb4
tff(fact_7394_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb4
tff(fact_7395_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb4
tff(fact_7396_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_7397_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_7398_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le72135733267957522d_enat(B,A)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ).

% max.absorb3
tff(fact_7399_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( aa_real_real(ord_max_real(A),B) = A ) ) ).

% max.absorb3
tff(fact_7400_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb3
tff(fact_7401_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb3
tff(fact_7402_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_7403_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_7404_max__bot2,axiom,
    ! [X3: set_real] : ord_max_set_real(X3,bot_bot_set_real) = X3 ).

% max_bot2
tff(fact_7405_max__bot2,axiom,
    ! [X3: set_bool] : ord_max_set_bool(X3,bot_bot_set_bool) = X3 ).

% max_bot2
tff(fact_7406_max__bot2,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X3),bot_bot_set_nat) = X3 ).

% max_bot2
tff(fact_7407_max__bot2,axiom,
    ! [X3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X3),bot_bot_set_int) = X3 ).

% max_bot2
tff(fact_7408_max__bot2,axiom,
    ! [X3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),bot_bot_nat) = X3 ).

% max_bot2
tff(fact_7409_max__bot2,axiom,
    ! [X3: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),bot_bo4199563552545308370d_enat) = X3 ).

% max_bot2
tff(fact_7410_max__bot,axiom,
    ! [X3: set_real] : ord_max_set_real(bot_bot_set_real,X3) = X3 ).

% max_bot
tff(fact_7411_max__bot,axiom,
    ! [X3: set_bool] : ord_max_set_bool(bot_bot_set_bool,X3) = X3 ).

% max_bot
tff(fact_7412_max__bot,axiom,
    ! [X3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,bot_bot_set_nat),X3) = X3 ).

% max_bot
tff(fact_7413_max__bot,axiom,
    ! [X3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,bot_bot_set_int),X3) = X3 ).

% max_bot
tff(fact_7414_max__bot,axiom,
    ! [X3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X3) = X3 ).

% max_bot
tff(fact_7415_max__bot,axiom,
    ! [X3: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(bot_bo4199563552545308370d_enat),X3) = X3 ).

% max_bot
tff(fact_7416_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).

% max_Suc_Suc
tff(fact_7417_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_7418_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_7419_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_7420_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_7421_max__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

% max_0L
tff(fact_7422_max__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

% max_0R
tff(fact_7423_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_7424_max__0__1_I1_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_7425_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_7426_max__0__1_I1_J,axiom,
    aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_7427_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_7428_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_7429_max__0__1_I2_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_7430_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_7431_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_7432_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_7433_Max__insert,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( lattic4901227151466704046nteger(insert_Code_integer(X3,A2)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(X3),lattic4901227151466704046nteger(A2)) ) ) ) ).

% Max_insert
tff(fact_7434_Max__insert,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( lattic4275903605611617917x_real(aa_set_real_set_real(insert_real(X3),A2)) = aa_real_real(ord_max_real(X3),lattic4275903605611617917x_real(A2)) ) ) ) ).

% Max_insert
tff(fact_7435_Max__insert,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( pp(aa_set_bool_bool(lattic7308977977949639686x_bool,aa_set_bool_set_bool(insert_bool(X3),A2)))
        <=> pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_max_bool,X3),aa_set_bool_bool(lattic7308977977949639686x_bool,A2))) ) ) ) ).

% Max_insert
tff(fact_7436_Max__insert,axiom,
    ! [A2: set_Extended_enat,X3: extended_enat] :
      ( finite4001608067531595151d_enat(A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( lattic921264341876707157d_enat(insert_Extended_enat(X3,A2)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),lattic921264341876707157d_enat(A2)) ) ) ) ).

% Max_insert
tff(fact_7437_Max__insert,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(X3),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)) ) ) ) ).

% Max_insert
tff(fact_7438_Max__insert,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(insert_int(X3),A2)) = aa_int_int(aa_int_fun_int_int(ord_max_int,X3),aa_set_int_int(lattic8263393255366662781ax_int,A2)) ) ) ) ).

% Max_insert
tff(fact_7439_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_7440_of__nat__max,axiom,
    ! [X3: nat,Y: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(semiri4216267220026989637d_enat(X3)),semiri4216267220026989637d_enat(Y)) ).

% of_nat_max
tff(fact_7441_of__nat__max,axiom,
    ! [X3: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)) = aa_real_real(ord_max_real(aa_nat_real(semiri5074537144036343181t_real,X3)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_7442_of__nat__max,axiom,
    ! [X3: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_7443_of__nat__max,axiom,
    ! [X3: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_7444_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(C,B)
     => ord_le2932123472753598470d_enat(C,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ).

% max.coboundedI2
tff(fact_7445_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ) ).

% max.coboundedI2
tff(fact_7446_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI2
tff(fact_7447_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI2
tff(fact_7448_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI2
tff(fact_7449_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(C,A)
     => ord_le2932123472753598470d_enat(C,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ).

% max.coboundedI1
tff(fact_7450_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ) ).

% max.coboundedI1
tff(fact_7451_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.coboundedI1
tff(fact_7452_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI1
tff(fact_7453_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI1
tff(fact_7454_max_Oabsorb__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(A,B)
    <=> ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_7455_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_7456_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_7457_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7458_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_7459_max_Oabsorb__iff1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
    <=> ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_7460_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_7461_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_7462_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7463_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_7464_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X3: extended_enat,Y: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),Y))
    <=> ( ord_le2932123472753598470d_enat(Z,X3)
        | ord_le2932123472753598470d_enat(Z,Y) ) ) ).

% le_max_iff_disj
tff(fact_7465_le__max__iff__disj,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X3),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X3))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_7466_le__max__iff__disj,axiom,
    ! [Z: num,X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X3))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_7467_le__max__iff__disj,axiom,
    ! [Z: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X3))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_7468_le__max__iff__disj,axiom,
    ! [Z: int,X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X3))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) ) ) ).

% le_max_iff_disj
tff(fact_7469_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ord_le2932123472753598470d_enat(B,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ).

% max.cobounded2
tff(fact_7470_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ).

% max.cobounded2
tff(fact_7471_max_Ocobounded2,axiom,
    ! [B: num,A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded2
tff(fact_7472_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).

% max.cobounded2
tff(fact_7473_max_Ocobounded2,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).

% max.cobounded2
tff(fact_7474_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ord_le2932123472753598470d_enat(A,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ).

% max.cobounded1
tff(fact_7475_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ).

% max.cobounded1
tff(fact_7476_max_Ocobounded1,axiom,
    ! [A: num,B: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).

% max.cobounded1
tff(fact_7477_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).

% max.cobounded1
tff(fact_7478_max_Ocobounded1,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).

% max.cobounded1
tff(fact_7479_max_Oorder__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
    <=> ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) ) ) ).

% max.order_iff
tff(fact_7480_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.order_iff
tff(fact_7481_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.order_iff
tff(fact_7482_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7483_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_7484_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
     => ( ord_le2932123472753598470d_enat(C,A)
       => ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C),A) ) ) ).

% max.boundedI
tff(fact_7485_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_7486_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_7487_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_7488_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_7489_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C),A)
     => ~ ( ord_le2932123472753598470d_enat(B,A)
         => ~ ord_le2932123472753598470d_enat(C,A) ) ) ).

% max.boundedE
tff(fact_7490_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% max.boundedE
tff(fact_7491_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% max.boundedE
tff(fact_7492_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.boundedE
tff(fact_7493_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% max.boundedE
tff(fact_7494_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) )
     => ord_le2932123472753598470d_enat(B,A) ) ).

% max.orderI
tff(fact_7495_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% max.orderI
tff(fact_7496_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% max.orderI
tff(fact_7497_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% max.orderI
tff(fact_7498_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% max.orderI
tff(fact_7499_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
     => ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) ) ) ).

% max.orderE
tff(fact_7500_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.orderE
tff(fact_7501_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.orderE
tff(fact_7502_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_7503_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_7504_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(C,A)
     => ( ord_le2932123472753598470d_enat(D,B)
       => ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(C),D),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ) ).

% max.mono
tff(fact_7505_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,C),D)),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ) ) ).

% max.mono
tff(fact_7506_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,D),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ) ).

% max.mono
tff(fact_7507_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ) ).

% max.mono
tff(fact_7508_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ) ).

% max.mono
tff(fact_7509_max__absorb2,axiom,
    ! [X3: extended_enat,Y: extended_enat] :
      ( ord_le2932123472753598470d_enat(X3,Y)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_7510_max__absorb2,axiom,
    ! [X3: set_int,Y: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X3),Y))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_7511_max__absorb2,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_7512_max__absorb2,axiom,
    ! [X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Y))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_7513_max__absorb2,axiom,
    ! [X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_7514_max__absorb2,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y) = Y ) ) ).

% max_absorb2
tff(fact_7515_max__absorb1,axiom,
    ! [Y: extended_enat,X3: extended_enat] :
      ( ord_le2932123472753598470d_enat(Y,X3)
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_7516_max__absorb1,axiom,
    ! [Y: set_int,X3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(Y),X3))
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_7517_max__absorb1,axiom,
    ! [Y: rat,X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X3))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_7518_max__absorb1,axiom,
    ! [Y: num,X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X3))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_7519_max__absorb1,axiom,
    ! [Y: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_7520_max__absorb1,axiom,
    ! [Y: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y) = X3 ) ) ).

% max_absorb1
tff(fact_7521_max__def,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat(A,B)
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) )
      & ( ~ ord_le2932123472753598470d_enat(A,B)
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ) ).

% max_def
tff(fact_7522_max__def,axiom,
    ! [A: set_int,B: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,A),B) = B ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(A),B))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,A),B) = A ) ) ) ).

% max_def
tff(fact_7523_max__def,axiom,
    ! [A: rat,B: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ) ).

% max_def
tff(fact_7524_max__def,axiom,
    ! [A: num,B: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ) ).

% max_def
tff(fact_7525_max__def,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ) ).

% max_def
tff(fact_7526_max__def,axiom,
    ! [A: int,B: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ) ).

% max_def
tff(fact_7527_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X3: extended_enat,Y: extended_enat] :
      ( ord_le72135733267957522d_enat(Z,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),Y))
    <=> ( ord_le72135733267957522d_enat(Z,X3)
        | ord_le72135733267957522d_enat(Z,Y) ) ) ).

% less_max_iff_disj
tff(fact_7528_less__max__iff__disj,axiom,
    ! [Z: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(ord_max_real(X3),Y)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X3))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_7529_less__max__iff__disj,axiom,
    ! [Z: rat,X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X3),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X3))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_7530_less__max__iff__disj,axiom,
    ! [Z: num,X3: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X3))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_7531_less__max__iff__disj,axiom,
    ! [Z: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X3))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_7532_less__max__iff__disj,axiom,
    ! [Z: int,X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X3))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_7533_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C),A)
     => ~ ( ord_le72135733267957522d_enat(B,A)
         => ~ ord_le72135733267957522d_enat(C,A) ) ) ).

% max.strict_boundedE
tff(fact_7534_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ord_max_real(B),C)),A))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_7535_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_7536_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_7537_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_7538_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_7539_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le72135733267957522d_enat(B,A)
    <=> ( ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7540_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( ( A = aa_real_real(ord_max_real(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7541_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7542_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7543_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_7544_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7545_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ord_le72135733267957522d_enat(C,A)
     => ord_le72135733267957522d_enat(C,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_7546_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(ord_max_real(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_7547_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_7548_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_7549_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_7550_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_7551_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ord_le72135733267957522d_enat(C,B)
     => ord_le72135733267957522d_enat(C,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_7552_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(ord_max_real(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_7553_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_7554_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_7555_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_7556_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_7557_max__add__distrib__left,axiom,
    ! [X3: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ord_max_real(X3),Y)),Z) = aa_real_real(ord_max_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_7558_max__add__distrib__left,axiom,
    ! [X3: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X3),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z)) ).

% max_add_distrib_left
tff(fact_7559_max__add__distrib__left,axiom,
    ! [X3: 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,X3),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_7560_max__add__distrib__left,axiom,
    ! [X3: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_7561_max__add__distrib__right,axiom,
    ! [X3: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),aa_real_real(ord_max_real(Y),Z)) = aa_real_real(ord_max_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Z)) ).

% max_add_distrib_right
tff(fact_7562_max__add__distrib__right,axiom,
    ! [X3: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Z)) ).

% max_add_distrib_right
tff(fact_7563_max__add__distrib__right,axiom,
    ! [X3: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),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,X3),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Z)) ).

% max_add_distrib_right
tff(fact_7564_max__add__distrib__right,axiom,
    ! [X3: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Z)) ).

% max_add_distrib_right
tff(fact_7565_max__diff__distrib__left,axiom,
    ! [X3: real,Y: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(ord_max_real(X3),Y)),Z) = aa_real_real(ord_max_real(aa_real_real(minus_minus_real(X3),Z)),aa_real_real(minus_minus_real(Y),Z)) ).

% max_diff_distrib_left
tff(fact_7566_max__diff__distrib__left,axiom,
    ! [X3: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X3),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),Z)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),Z)) ).

% max_diff_distrib_left
tff(fact_7567_max__diff__distrib__left,axiom,
    ! [X3: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_7568_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q4)) ).

% nat_add_max_left
tff(fact_7569_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ).

% nat_add_max_right
tff(fact_7570_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ).

% nat_mult_max_left
tff(fact_7571_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_max_right
tff(fact_7572_max__def__raw,axiom,
    ! [X5: extended_enat,Xa2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat(X5,Xa2)
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X5),Xa2) = Xa2 ) )
      & ( ~ ord_le2932123472753598470d_enat(X5,Xa2)
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_7573_max__def__raw,axiom,
    ! [X5: set_int,Xa2: set_int] :
      ( ( pp(aa_set_int_bool(ord_less_eq_set_int(X5),Xa2))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_set_int_bool(ord_less_eq_set_int(X5),Xa2))
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_7574_max__def__raw,axiom,
    ! [X5: rat,Xa2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Xa2))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X5),Xa2))
       => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_7575_max__def__raw,axiom,
    ! [X5: num,Xa2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Xa2))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X5),Xa2))
       => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_7576_max__def__raw,axiom,
    ! [X5: nat,Xa2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Xa2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_7577_max__def__raw,axiom,
    ! [X5: int,Xa2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Xa2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X5),Xa2) = Xa2 ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),Xa2))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X5),Xa2) = X5 ) ) ) ).

% max_def_raw
tff(fact_7578_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).

% nat_minus_add_max
tff(fact_7579_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_cf,aTP_Lamm_cg) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_7580_Sup__insert__finite,axiom,
    ! [S3: set_Extended_enat,X3: extended_enat] :
      ( finite4001608067531595151d_enat(S3)
     => ( ( ( S3 = bot_bo7653980558646680370d_enat )
         => ( comple4398354569131411667d_enat(insert_Extended_enat(X3,S3)) = X3 ) )
        & ( ( S3 != bot_bo7653980558646680370d_enat )
         => ( comple4398354569131411667d_enat(insert_Extended_enat(X3,S3)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),comple4398354569131411667d_enat(S3)) ) ) ) ) ).

% Sup_insert_finite
tff(fact_7581_Sup__insert__finite,axiom,
    ! [S3: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( ( S3 = bot_bot_set_int )
         => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(insert_int(X3),S3)) = X3 ) )
        & ( ( S3 != bot_bot_set_int )
         => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(insert_int(X3),S3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,X3),aa_set_int_int(complete_Sup_Sup_int,S3)) ) ) ) ) ).

% Sup_insert_finite
tff(fact_7582_Sup__insert__finite,axiom,
    ! [S3: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( ( S3 = bot_bot_set_nat )
         => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(insert_nat(X3),S3)) = X3 ) )
        & ( ( S3 != bot_bot_set_nat )
         => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(insert_nat(X3),S3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),aa_set_nat_nat(complete_Sup_Sup_nat,S3)) ) ) ) ) ).

% Sup_insert_finite
tff(fact_7583_Sup__insert__finite,axiom,
    ! [S3: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( ( S3 = bot_bot_set_real )
         => ( comple1385675409528146559p_real(aa_set_real_set_real(insert_real(X3),S3)) = X3 ) )
        & ( ( S3 != bot_bot_set_real )
         => ( comple1385675409528146559p_real(aa_set_real_set_real(insert_real(X3),S3)) = aa_real_real(ord_max_real(X3),comple1385675409528146559p_real(S3)) ) ) ) ) ).

% Sup_insert_finite
tff(fact_7584_hom__Max__commute,axiom,
    ! [H: fun_Co8499238673658658103nteger,N7: set_Code_integer] :
      ( ! [X: code_integer,Y3: code_integer] : aa_Cod1783974652533396728nteger(H,aa_Cod1783974652533396728nteger(ord_max_Code_integer(X),Y3)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_Cod1783974652533396728nteger(H,X)),aa_Cod1783974652533396728nteger(H,Y3))
     => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,N7))
       => ( ( N7 != bot_bo3990330152332043303nteger )
         => ( aa_Cod1783974652533396728nteger(H,lattic4901227151466704046nteger(N7)) = lattic4901227151466704046nteger(image_4470545334726330049nteger(H,N7)) ) ) ) ) ).

% hom_Max_commute
tff(fact_7585_hom__Max__commute,axiom,
    ! [H: fun_real_real,N7: set_real] :
      ( ! [X: real,Y3: real] : aa_real_real(H,aa_real_real(ord_max_real(X),Y3)) = aa_real_real(ord_max_real(aa_real_real(H,X)),aa_real_real(H,Y3))
     => ( pp(aa_set_real_bool(finite_finite_real,N7))
       => ( ( N7 != bot_bot_set_real )
         => ( aa_real_real(H,lattic4275903605611617917x_real(N7)) = lattic4275903605611617917x_real(image_real_real(H,N7)) ) ) ) ) ).

% hom_Max_commute
tff(fact_7586_hom__Max__commute,axiom,
    ! [H: fun_bool_bool,N7: set_bool] :
      ( ! [X: bool,Y3: bool] :
          ( pp(aa_bool_bool(H,aa_bool_bool(aa_boo8970175072276269406l_bool(ord_max_bool,X),Y3)))
        <=> pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_max_bool,aa_bool_bool(H,X)),aa_bool_bool(H,Y3))) )
     => ( pp(aa_set_bool_bool(finite_finite_bool,N7))
       => ( ( N7 != bot_bot_set_bool )
         => ( pp(aa_bool_bool(H,aa_set_bool_bool(lattic7308977977949639686x_bool,N7)))
          <=> pp(aa_set_bool_bool(lattic7308977977949639686x_bool,image_bool_bool(H,N7))) ) ) ) ) ).

% hom_Max_commute
tff(fact_7587_hom__Max__commute,axiom,
    ! [H: fun_Ex5852502975177633731d_enat,N7: set_Extended_enat] :
      ( ! [X: extended_enat,Y3: extended_enat] : aa_Ext7969882492699152700d_enat(H,aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Y3)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(aa_Ext7969882492699152700d_enat(H,X)),aa_Ext7969882492699152700d_enat(H,Y3))
     => ( finite4001608067531595151d_enat(N7)
       => ( ( N7 != bot_bo7653980558646680370d_enat )
         => ( aa_Ext7969882492699152700d_enat(H,lattic921264341876707157d_enat(N7)) = lattic921264341876707157d_enat(image_80655429650038917d_enat(H,N7)) ) ) ) ) ).

% hom_Max_commute
tff(fact_7588_hom__Max__commute,axiom,
    ! [H: fun_nat_nat,N7: set_nat] :
      ( ! [X: nat,Y3: nat] : aa_nat_nat(H,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(H,X)),aa_nat_nat(H,Y3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,N7))
       => ( ( N7 != bot_bot_set_nat )
         => ( aa_nat_nat(H,aa_set_nat_nat(lattic8265883725875713057ax_nat,N7)) = aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(image_nat_nat(H),N7)) ) ) ) ) ).

% hom_Max_commute
tff(fact_7589_hom__Max__commute,axiom,
    ! [H: fun_int_int,N7: set_int] :
      ( ! [X: int,Y3: int] : aa_int_int(H,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(H,X)),aa_int_int(H,Y3))
     => ( pp(aa_set_int_bool(finite_finite_int,N7))
       => ( ( N7 != bot_bot_set_int )
         => ( aa_int_int(H,aa_set_int_int(lattic8263393255366662781ax_int,N7)) = aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(image_int_int(H),N7)) ) ) ) ) ).

% hom_Max_commute
tff(fact_7590_Max_Osubset,axiom,
    ! [A2: set_Code_integer,B3: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( B3 != bot_bo3990330152332043303nteger )
       => ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(B3),A2))
         => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(lattic4901227151466704046nteger(B3)),lattic4901227151466704046nteger(A2)) = lattic4901227151466704046nteger(A2) ) ) ) ) ).

% Max.subset
tff(fact_7591_Max_Osubset,axiom,
    ! [A2: set_real,B3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( B3 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B3),A2))
         => ( aa_real_real(ord_max_real(lattic4275903605611617917x_real(B3)),lattic4275903605611617917x_real(A2)) = lattic4275903605611617917x_real(A2) ) ) ) ) ).

% Max.subset
tff(fact_7592_Max_Osubset,axiom,
    ! [A2: set_bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( B3 != bot_bot_set_bool )
       => ( pp(aa_set_bool_bool(ord_less_eq_set_bool(B3),A2))
         => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_max_bool,aa_set_bool_bool(lattic7308977977949639686x_bool,B3)),aa_set_bool_bool(lattic7308977977949639686x_bool,A2)))
          <=> pp(aa_set_bool_bool(lattic7308977977949639686x_bool,A2)) ) ) ) ) ).

% Max.subset
tff(fact_7593_Max_Osubset,axiom,
    ! [A2: set_Extended_enat,B3: set_Extended_enat] :
      ( finite4001608067531595151d_enat(A2)
     => ( ( B3 != bot_bo7653980558646680370d_enat )
       => ( ord_le7203529160286727270d_enat(B3,A2)
         => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(lattic921264341876707157d_enat(B3)),lattic921264341876707157d_enat(A2)) = lattic921264341876707157d_enat(A2) ) ) ) ) ).

% Max.subset
tff(fact_7594_Max_Osubset,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( B3 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,B3),A2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,B3)),aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)) = aa_set_nat_nat(lattic8265883725875713057ax_nat,A2) ) ) ) ) ).

% Max.subset
tff(fact_7595_Max_Osubset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( B3 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B3),A2))
         => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_set_int_int(lattic8263393255366662781ax_int,B3)),aa_set_int_int(lattic8263393255366662781ax_int,A2)) = aa_set_int_int(lattic8263393255366662781ax_int,A2) ) ) ) ) ).

% Max.subset
tff(fact_7596_Max_Oclosed,axiom,
    ! [A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ! [X: code_integer,Y3: code_integer] : pp(aa_set1374477347934578310r_bool(member_Code_integer(aa_Cod1783974652533396728nteger(ord_max_Code_integer(X),Y3)),insert_Code_integer(X,insert_Code_integer(Y3,bot_bo3990330152332043303nteger))))
         => pp(aa_set1374477347934578310r_bool(member_Code_integer(lattic4901227151466704046nteger(A2)),A2)) ) ) ) ).

% Max.closed
tff(fact_7597_Max_Oclosed,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X: real,Y3: real] : pp(aa_set_real_bool(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))))
         => pp(aa_set_real_bool(member_real(lattic4275903605611617917x_real(A2)),A2)) ) ) ) ).

% Max.closed
tff(fact_7598_Max_Oclosed,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( ! [X: bool,Y3: bool] : pp(aa_set_bool_bool(member_bool(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_max_bool,X),Y3)),aa_set_bool_set_bool(insert_bool(X),aa_set_bool_set_bool(insert_bool(Y3),bot_bot_set_bool))))
         => pp(aa_set_bool_bool(member_bool(aa_set_bool_bool(lattic7308977977949639686x_bool,A2)),A2)) ) ) ) ).

% Max.closed
tff(fact_7599_Max_Oclosed,axiom,
    ! [A2: set_Extended_enat] :
      ( finite4001608067531595151d_enat(A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X: extended_enat,Y3: extended_enat] : member_Extended_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Y3),insert_Extended_enat(X,insert_Extended_enat(Y3,bot_bo7653980558646680370d_enat)))
         => member_Extended_enat(lattic921264341876707157d_enat(A2),A2) ) ) ) ).

% Max.closed
tff(fact_7600_Max_Oclosed,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X: nat,Y3: nat] : pp(aa_set_nat_bool(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))))
         => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)),A2)) ) ) ) ).

% Max.closed
tff(fact_7601_Max_Oclosed,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X: int,Y3: int] : pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_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))))
         => pp(aa_set_int_bool(member_int(aa_set_int_int(lattic8263393255366662781ax_int,A2)),A2)) ) ) ) ).

% Max.closed
tff(fact_7602_Max_Oinsert__not__elem,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A2))
       => ( ( A2 != bot_bo3990330152332043303nteger )
         => ( lattic4901227151466704046nteger(insert_Code_integer(X3,A2)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(X3),lattic4901227151466704046nteger(A2)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_7603_Max_Oinsert__not__elem,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X3),A2))
       => ( ( A2 != bot_bot_set_real )
         => ( lattic4275903605611617917x_real(aa_set_real_set_real(insert_real(X3),A2)) = aa_real_real(ord_max_real(X3),lattic4275903605611617917x_real(A2)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_7604_Max_Oinsert__not__elem,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ~ pp(aa_set_bool_bool(member_bool(X3),A2))
       => ( ( A2 != bot_bot_set_bool )
         => ( pp(aa_set_bool_bool(lattic7308977977949639686x_bool,aa_set_bool_set_bool(insert_bool(X3),A2)))
          <=> pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_max_bool,X3),aa_set_bool_bool(lattic7308977977949639686x_bool,A2))) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_7605_Max_Oinsert__not__elem,axiom,
    ! [A2: set_Extended_enat,X3: extended_enat] :
      ( finite4001608067531595151d_enat(A2)
     => ( ~ member_Extended_enat(X3,A2)
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => ( lattic921264341876707157d_enat(insert_Extended_enat(X3,A2)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),lattic921264341876707157d_enat(A2)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_7606_Max_Oinsert__not__elem,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X3),A2))
       => ( ( A2 != bot_bot_set_nat )
         => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(X3),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_7607_Max_Oinsert__not__elem,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X3),A2))
       => ( ( A2 != bot_bot_set_int )
         => ( aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(insert_int(X3),A2)) = aa_int_int(aa_int_fun_int_int(ord_max_int,X3),aa_set_int_int(lattic8263393255366662781ax_int,A2)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_7608_Max_Ounion,axiom,
    ! [A2: set_Code_integer,B3: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,B3))
         => ( ( B3 != bot_bo3990330152332043303nteger )
           => ( lattic4901227151466704046nteger(sup_su848401254843788991nteger(A2,B3)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(lattic4901227151466704046nteger(A2)),lattic4901227151466704046nteger(B3)) ) ) ) ) ) ).

% Max.union
tff(fact_7609_Max_Ounion,axiom,
    ! [A2: set_real,B3: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( pp(aa_set_real_bool(finite_finite_real,B3))
         => ( ( B3 != bot_bot_set_real )
           => ( lattic4275903605611617917x_real(aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,A2),B3)) = aa_real_real(ord_max_real(lattic4275903605611617917x_real(A2)),lattic4275903605611617917x_real(B3)) ) ) ) ) ) ).

% Max.union
tff(fact_7610_Max_Ounion,axiom,
    ! [A2: set_bool,B3: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( A2 != bot_bot_set_bool )
       => ( pp(aa_set_bool_bool(finite_finite_bool,B3))
         => ( ( B3 != bot_bot_set_bool )
           => ( pp(aa_set_bool_bool(lattic7308977977949639686x_bool,aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),B3)))
            <=> pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_max_bool,aa_set_bool_bool(lattic7308977977949639686x_bool,A2)),aa_set_bool_bool(lattic7308977977949639686x_bool,B3))) ) ) ) ) ) ).

% Max.union
tff(fact_7611_Max_Ounion,axiom,
    ! [A2: set_Extended_enat,B3: set_Extended_enat] :
      ( finite4001608067531595151d_enat(A2)
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( finite4001608067531595151d_enat(B3)
         => ( ( B3 != bot_bo7653980558646680370d_enat )
           => ( lattic921264341876707157d_enat(sup_su4489774667511045786d_enat(A2,B3)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(lattic921264341876707157d_enat(A2)),lattic921264341876707157d_enat(B3)) ) ) ) ) ) ).

% Max.union
tff(fact_7612_Max_Ounion,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
         => ( ( B3 != bot_bot_set_nat )
           => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A2)),aa_set_nat_nat(lattic8265883725875713057ax_nat,B3)) ) ) ) ) ) ).

% Max.union
tff(fact_7613_Max_Ounion,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(finite_finite_int,B3))
         => ( ( B3 != bot_bot_set_int )
           => ( aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_set_int_int(lattic8263393255366662781ax_int,A2)),aa_set_int_int(lattic8263393255366662781ax_int,B3)) ) ) ) ) ) ).

% Max.union
tff(fact_7614_conj__subset__def,axiom,
    ! [A2: set_complex,P: fun_complex_bool,Q: fun_complex_bool] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_cp(P),Q))))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),collect_complex(P)))
        & pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),collect_complex(Q))) ) ) ).

% conj_subset_def
tff(fact_7615_conj__subset__def,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ord_le6045566169113846134st_nat(A2,collect_list_nat(aa_fun6009467934925565500t_bool(aTP_Lamm_cl(P),Q)))
    <=> ( ord_le6045566169113846134st_nat(A2,collect_list_nat(P))
        & ord_le6045566169113846134st_nat(A2,collect_list_nat(Q)) ) ) ).

% conj_subset_def
tff(fact_7616_conj__subset__def,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),collect_set_nat(aa_fun3565389549526402364t_bool(aTP_Lamm_cm(P),Q))))
    <=> ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),collect_set_nat(P)))
        & pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(A2),collect_set_nat(Q))) ) ) ).

% conj_subset_def
tff(fact_7617_conj__subset__def,axiom,
    ! [A2: set_nat,P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_cn(P),Q))))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),collect_nat(P)))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),collect_nat(Q))) ) ) ).

% conj_subset_def
tff(fact_7618_conj__subset__def,axiom,
    ! [A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_co(P),Q))))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),collect_int(P)))
        & pp(aa_set_int_bool(ord_less_eq_set_int(A2),collect_int(Q))) ) ) ).

% conj_subset_def
tff(fact_7619_Max_Oremove,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A2))
       => ( ( ( minus_2355218937544613996nteger(A2,insert_Code_integer(X3,bot_bo3990330152332043303nteger)) = bot_bo3990330152332043303nteger )
           => ( lattic4901227151466704046nteger(A2) = X3 ) )
          & ( ( minus_2355218937544613996nteger(A2,insert_Code_integer(X3,bot_bo3990330152332043303nteger)) != bot_bo3990330152332043303nteger )
           => ( lattic4901227151466704046nteger(A2) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(X3),lattic4901227151466704046nteger(minus_2355218937544613996nteger(A2,insert_Code_integer(X3,bot_bo3990330152332043303nteger)))) ) ) ) ) ) ).

% Max.remove
tff(fact_7620_Max_Oremove,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(member_real(X3),A2))
       => ( ( ( minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) = bot_bot_set_real )
           => ( lattic4275903605611617917x_real(A2) = X3 ) )
          & ( ( minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) != bot_bot_set_real )
           => ( lattic4275903605611617917x_real(A2) = aa_real_real(ord_max_real(X3),lattic4275903605611617917x_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))) ) ) ) ) ) ).

% Max.remove
tff(fact_7621_Max_Oremove,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( pp(aa_set_bool_bool(member_bool(X3),A2))
       => ( pp(aa_set_bool_bool(lattic7308977977949639686x_bool,A2))
        <=> ( ( ( minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)) = bot_bot_set_bool )
             => pp(X3) )
            & ( ( minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)) != bot_bot_set_bool )
             => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_max_bool,X3),aa_set_bool_bool(lattic7308977977949639686x_bool,minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))))) ) ) ) ) ) ).

% Max.remove
tff(fact_7622_Max_Oremove,axiom,
    ! [A2: set_Extended_enat,X3: extended_enat] :
      ( finite4001608067531595151d_enat(A2)
     => ( member_Extended_enat(X3,A2)
       => ( ( ( minus_925952699566721837d_enat(A2,insert_Extended_enat(X3,bot_bo7653980558646680370d_enat)) = bot_bo7653980558646680370d_enat )
           => ( lattic921264341876707157d_enat(A2) = X3 ) )
          & ( ( minus_925952699566721837d_enat(A2,insert_Extended_enat(X3,bot_bo7653980558646680370d_enat)) != bot_bo7653980558646680370d_enat )
           => ( lattic921264341876707157d_enat(A2) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),lattic921264341876707157d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(X3,bot_bo7653980558646680370d_enat)))) ) ) ) ) ) ).

% Max.remove
tff(fact_7623_Max_Oremove,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(X3),A2))
       => ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) = bot_bot_set_nat )
           => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A2) = X3 ) )
          & ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) != bot_bot_set_nat )
           => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)))) ) ) ) ) ) ).

% Max.remove
tff(fact_7624_Max_Oremove,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X3),A2))
       => ( ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) = bot_bot_set_int )
           => ( aa_set_int_int(lattic8263393255366662781ax_int,A2) = X3 ) )
          & ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) != bot_bot_set_int )
           => ( aa_set_int_int(lattic8263393255366662781ax_int,A2) = aa_int_int(aa_int_fun_int_int(ord_max_int,X3),aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)))) ) ) ) ) ) ).

% Max.remove
tff(fact_7625_Max_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,X3: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( ( minus_2355218937544613996nteger(A2,insert_Code_integer(X3,bot_bo3990330152332043303nteger)) = bot_bo3990330152332043303nteger )
         => ( lattic4901227151466704046nteger(insert_Code_integer(X3,A2)) = X3 ) )
        & ( ( minus_2355218937544613996nteger(A2,insert_Code_integer(X3,bot_bo3990330152332043303nteger)) != bot_bo3990330152332043303nteger )
         => ( lattic4901227151466704046nteger(insert_Code_integer(X3,A2)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(X3),lattic4901227151466704046nteger(minus_2355218937544613996nteger(A2,insert_Code_integer(X3,bot_bo3990330152332043303nteger)))) ) ) ) ) ).

% Max.insert_remove
tff(fact_7626_Max_Oinsert__remove,axiom,
    ! [A2: set_real,X3: real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( ( minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) = bot_bot_set_real )
         => ( lattic4275903605611617917x_real(aa_set_real_set_real(insert_real(X3),A2)) = X3 ) )
        & ( ( minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)) != bot_bot_set_real )
         => ( lattic4275903605611617917x_real(aa_set_real_set_real(insert_real(X3),A2)) = aa_real_real(ord_max_real(X3),lattic4275903605611617917x_real(minus_minus_set_real(A2,aa_set_real_set_real(insert_real(X3),bot_bot_set_real)))) ) ) ) ) ).

% Max.insert_remove
tff(fact_7627_Max_Oinsert__remove,axiom,
    ! [A2: set_bool,X3: bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( pp(aa_set_bool_bool(lattic7308977977949639686x_bool,aa_set_bool_set_bool(insert_bool(X3),A2)))
      <=> ( ( ( minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)) = bot_bot_set_bool )
           => pp(X3) )
          & ( ( minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool)) != bot_bot_set_bool )
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_max_bool,X3),aa_set_bool_bool(lattic7308977977949639686x_bool,minus_minus_set_bool(A2,aa_set_bool_set_bool(insert_bool(X3),bot_bot_set_bool))))) ) ) ) ) ).

% Max.insert_remove
tff(fact_7628_Max_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,X3: extended_enat] :
      ( finite4001608067531595151d_enat(A2)
     => ( ( ( minus_925952699566721837d_enat(A2,insert_Extended_enat(X3,bot_bo7653980558646680370d_enat)) = bot_bo7653980558646680370d_enat )
         => ( lattic921264341876707157d_enat(insert_Extended_enat(X3,A2)) = X3 ) )
        & ( ( minus_925952699566721837d_enat(A2,insert_Extended_enat(X3,bot_bo7653980558646680370d_enat)) != bot_bo7653980558646680370d_enat )
         => ( lattic921264341876707157d_enat(insert_Extended_enat(X3,A2)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X3),lattic921264341876707157d_enat(minus_925952699566721837d_enat(A2,insert_Extended_enat(X3,bot_bo7653980558646680370d_enat)))) ) ) ) ) ).

% Max.insert_remove
tff(fact_7629_Max_Oinsert__remove,axiom,
    ! [A2: set_nat,X3: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) = bot_bot_set_nat )
         => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(X3),A2)) = X3 ) )
        & ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)) != bot_bot_set_nat )
         => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(X3),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)))) ) ) ) ) ).

% Max.insert_remove
tff(fact_7630_Max_Oinsert__remove,axiom,
    ! [A2: set_int,X3: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) = bot_bot_set_int )
         => ( aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(insert_int(X3),A2)) = X3 ) )
        & ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)) != bot_bot_set_int )
         => ( aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(insert_int(X3),A2)) = aa_int_int(aa_int_fun_int_int(ord_max_int,X3),aa_set_int_int(lattic8263393255366662781ax_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A2),aa_set_int_set_int(insert_int(X3),bot_bot_set_int)))) ) ) ) ) ).

% Max.insert_remove
tff(fact_7631_Sup__option__def,axiom,
    ! [A2: set_option_set_nat] :
      ( ( ( ( A2 = bot_bo5650944848895983264et_nat )
          | ( A2 = insert6281397740311767046et_nat(none_set_nat,bot_bo5650944848895983264et_nat) ) )
       => ( comple8455683388168444585et_nat(A2) = none_set_nat ) )
      & ( ~ ( ( A2 = bot_bo5650944848895983264et_nat )
            | ( A2 = insert6281397740311767046et_nat(none_set_nat,bot_bo5650944848895983264et_nat) ) )
       => ( comple8455683388168444585et_nat(A2) = aa_set6119337032432945050et_nat(some_set_nat,comple7399068483239264473et_nat(these_set_nat(A2))) ) ) ) ).

% Sup_option_def
tff(fact_7632_UN__constant__eq,axiom,
    ! [A: nat,A2: set_nat,F: fun_nat_set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),A2))
           => ( aa_nat_set_nat(F,X) = C ) )
       => ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)) = C ) ) ) ).

% UN_constant_eq
tff(fact_7633_UN__constant__eq,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VE9028255658833972685et_nat,C: set_nat] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),A2))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
           => ( aa_VEBT_VEBT_set_nat(F,X) = C ) )
       => ( comple7399068483239264473et_nat(image_6451421511446451829et_nat(F,A2)) = C ) ) ) ).

% UN_constant_eq
tff(fact_7634_UN__constant__eq,axiom,
    ! [A: real,A2: set_real,F: fun_real_set_nat,C: set_nat] :
      ( pp(aa_set_real_bool(member_real(A),A2))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),A2))
           => ( aa_real_set_nat(F,X) = C ) )
       => ( comple7399068483239264473et_nat(image_real_set_nat(F,A2)) = C ) ) ) ).

% UN_constant_eq
tff(fact_7635_UN__constant__eq,axiom,
    ! [A: int,A2: set_int,F: fun_int_set_nat,C: set_nat] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),A2))
           => ( aa_int_set_nat(F,X) = C ) )
       => ( comple7399068483239264473et_nat(image_int_set_nat(F,A2)) = C ) ) ) ).

% UN_constant_eq
tff(fact_7636_UN__constant__eq,axiom,
    ! [A: set_nat,A2: set_set_nat,F: fun_set_nat_set_nat,C: set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(A),A2))
     => ( ! [X: set_nat] :
            ( pp(aa_set_set_nat_bool(member_set_nat(X),A2))
           => ( aa_set_nat_set_nat(F,X) = C ) )
       => ( comple7399068483239264473et_nat(image_7916887816326733075et_nat(F,A2)) = C ) ) ) ).

% UN_constant_eq
tff(fact_7637_these__not__empty__eq,axiom,
    ! [B3: set_op4508134149509766951at_nat] :
      ( ( these_8061402112538453438at_nat(B3) != bot_bo2099793752762293965at_nat )
    <=> ( ( B3 != bot_bo3414888551220300179at_nat )
        & ( B3 != insert6842972033726296599at_nat(none_P5556105721700978146at_nat,bot_bo3414888551220300179at_nat) ) ) ) ).

% these_not_empty_eq
tff(fact_7638_these__not__empty__eq,axiom,
    ! [B3: set_option_num] :
      ( ( these_num(B3) != bot_bot_set_num )
    <=> ( ( B3 != bot_bo725063100934353204on_num )
        & ( B3 != insert_option_num(none_num,bot_bo725063100934353204on_num) ) ) ) ).

% these_not_empty_eq
tff(fact_7639_these__not__empty__eq,axiom,
    ! [B3: set_option_real] :
      ( ( these_real(B3) != bot_bot_set_real )
    <=> ( ( B3 != bot_bo677810143523618246n_real )
        & ( B3 != insert_option_real(none_real,bot_bo677810143523618246n_real) ) ) ) ).

% these_not_empty_eq
tff(fact_7640_these__not__empty__eq,axiom,
    ! [B3: set_option_bool] :
      ( ( these_bool(B3) != bot_bot_set_bool )
    <=> ( ( B3 != bot_bo4680648155223447721n_bool )
        & ( B3 != insert_option_bool(none_bool,bot_bo4680648155223447721n_bool) ) ) ) ).

% these_not_empty_eq
tff(fact_7641_these__not__empty__eq,axiom,
    ! [B3: set_option_nat] :
      ( ( these_nat(B3) != bot_bot_set_nat )
    <=> ( ( B3 != bot_bo5009843511495006442on_nat )
        & ( B3 != insert_option_nat(none_nat,bot_bo5009843511495006442on_nat) ) ) ) ).

% these_not_empty_eq
tff(fact_7642_these__not__empty__eq,axiom,
    ! [B3: set_option_int] :
      ( ( these_int(B3) != bot_bot_set_int )
    <=> ( ( B3 != bot_bo2519760366563279686on_int )
        & ( B3 != insert_option_int(none_int,bot_bo2519760366563279686on_int) ) ) ) ).

% these_not_empty_eq
tff(fact_7643_these__empty__eq,axiom,
    ! [B3: set_op4508134149509766951at_nat] :
      ( ( these_8061402112538453438at_nat(B3) = bot_bo2099793752762293965at_nat )
    <=> ( ( B3 = bot_bo3414888551220300179at_nat )
        | ( B3 = insert6842972033726296599at_nat(none_P5556105721700978146at_nat,bot_bo3414888551220300179at_nat) ) ) ) ).

% these_empty_eq
tff(fact_7644_these__empty__eq,axiom,
    ! [B3: set_option_num] :
      ( ( these_num(B3) = bot_bot_set_num )
    <=> ( ( B3 = bot_bo725063100934353204on_num )
        | ( B3 = insert_option_num(none_num,bot_bo725063100934353204on_num) ) ) ) ).

% these_empty_eq
tff(fact_7645_these__empty__eq,axiom,
    ! [B3: set_option_real] :
      ( ( these_real(B3) = bot_bot_set_real )
    <=> ( ( B3 = bot_bo677810143523618246n_real )
        | ( B3 = insert_option_real(none_real,bot_bo677810143523618246n_real) ) ) ) ).

% these_empty_eq
tff(fact_7646_these__empty__eq,axiom,
    ! [B3: set_option_bool] :
      ( ( these_bool(B3) = bot_bot_set_bool )
    <=> ( ( B3 = bot_bo4680648155223447721n_bool )
        | ( B3 = insert_option_bool(none_bool,bot_bo4680648155223447721n_bool) ) ) ) ).

% these_empty_eq
tff(fact_7647_these__empty__eq,axiom,
    ! [B3: set_option_nat] :
      ( ( these_nat(B3) = bot_bot_set_nat )
    <=> ( ( B3 = bot_bo5009843511495006442on_nat )
        | ( B3 = insert_option_nat(none_nat,bot_bo5009843511495006442on_nat) ) ) ) ).

% these_empty_eq
tff(fact_7648_these__empty__eq,axiom,
    ! [B3: set_option_int] :
      ( ( these_int(B3) = bot_bot_set_int )
    <=> ( ( B3 = bot_bo2519760366563279686on_int )
        | ( B3 = insert_option_int(none_int,bot_bo2519760366563279686on_int) ) ) ) ).

% these_empty_eq
tff(fact_7649_these__insert__Some,axiom,
    ! [X3: vEBT_VEBT,A2: set_option_VEBT_VEBT] : these_VEBT_VEBT(insert9052096497370831178T_VEBT(some_VEBT_VEBT(X3),A2)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),these_VEBT_VEBT(A2)) ).

% these_insert_Some
tff(fact_7650_these__insert__Some,axiom,
    ! [X3: int,A2: set_option_int] : these_int(insert_option_int(aa_int_option_int(some_int,X3),A2)) = aa_set_int_set_int(insert_int(X3),these_int(A2)) ).

% these_insert_Some
tff(fact_7651_these__insert__Some,axiom,
    ! [X3: bool,A2: set_option_bool] : these_bool(insert_option_bool(aa_bool_option_bool(some_bool,X3),A2)) = aa_set_bool_set_bool(insert_bool(X3),these_bool(A2)) ).

% these_insert_Some
tff(fact_7652_these__insert__Some,axiom,
    ! [X3: real,A2: set_option_real] : these_real(insert_option_real(aa_real_option_real(some_real,X3),A2)) = aa_set_real_set_real(insert_real(X3),these_real(A2)) ).

% these_insert_Some
tff(fact_7653_these__insert__Some,axiom,
    ! [X3: product_prod_nat_nat,A2: set_op4508134149509766951at_nat] : these_8061402112538453438at_nat(insert6842972033726296599at_nat(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X3),A2)) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X3),these_8061402112538453438at_nat(A2)) ).

% these_insert_Some
tff(fact_7654_these__insert__Some,axiom,
    ! [X3: nat,A2: set_option_nat] : these_nat(insert_option_nat(aa_nat_option_nat(some_nat,X3),A2)) = aa_set_nat_set_nat(insert_nat(X3),these_nat(A2)) ).

% these_insert_Some
tff(fact_7655_these__insert__Some,axiom,
    ! [X3: num,A2: set_option_num] : these_num(insert_option_num(aa_num_option_num(some_num,X3),A2)) = insert_num(X3,these_num(A2)) ).

% these_insert_Some
tff(fact_7656_these__image__Some__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : these_8061402112538453438at_nat(image_4198897800814241419at_nat(some_P7363390416028606310at_nat,A2)) = A2 ).

% these_image_Some_eq
tff(fact_7657_these__image__Some__eq,axiom,
    ! [A2: set_nat] : these_nat(image_nat_option_nat(some_nat,A2)) = A2 ).

% these_image_Some_eq
tff(fact_7658_these__image__Some__eq,axiom,
    ! [A2: set_num] : these_num(image_num_option_num(some_num,A2)) = A2 ).

% these_image_Some_eq
tff(fact_7659_these__empty,axiom,
    these_real(bot_bo677810143523618246n_real) = bot_bot_set_real ).

% these_empty
tff(fact_7660_these__empty,axiom,
    these_bool(bot_bo4680648155223447721n_bool) = bot_bot_set_bool ).

% these_empty
tff(fact_7661_these__empty,axiom,
    these_nat(bot_bo5009843511495006442on_nat) = bot_bot_set_nat ).

% these_empty
tff(fact_7662_these__empty,axiom,
    these_int(bot_bo2519760366563279686on_int) = bot_bot_set_int ).

% these_empty
tff(fact_7663_these__insert__None,axiom,
    ! [A2: set_option_nat] : these_nat(insert_option_nat(none_nat,A2)) = these_nat(A2) ).

% these_insert_None
tff(fact_7664_these__insert__None,axiom,
    ! [A2: set_op4508134149509766951at_nat] : these_8061402112538453438at_nat(insert6842972033726296599at_nat(none_P5556105721700978146at_nat,A2)) = these_8061402112538453438at_nat(A2) ).

% these_insert_None
tff(fact_7665_these__insert__None,axiom,
    ! [A2: set_option_num] : these_num(insert_option_num(none_num,A2)) = these_num(A2) ).

% these_insert_None
tff(fact_7666_in__these__eq,axiom,
    ! [X3: vEBT_VEBT,A2: set_option_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),these_VEBT_VEBT(A2)))
    <=> member2458453091852628771T_VEBT(some_VEBT_VEBT(X3),A2) ) ).

% in_these_eq
tff(fact_7667_in__these__eq,axiom,
    ! [X3: real,A2: set_option_real] :
      ( pp(aa_set_real_bool(member_real(X3),these_real(A2)))
    <=> member_option_real(aa_real_option_real(some_real,X3),A2) ) ).

% in_these_eq
tff(fact_7668_in__these__eq,axiom,
    ! [X3: int,A2: set_option_int] :
      ( pp(aa_set_int_bool(member_int(X3),these_int(A2)))
    <=> member_option_int(aa_int_option_int(some_int,X3),A2) ) ).

% in_these_eq
tff(fact_7669_in__these__eq,axiom,
    ! [X3: set_nat,A2: set_option_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),these_set_nat(A2)))
    <=> member8989860449721436141et_nat(aa_set6119337032432945050et_nat(some_set_nat,X3),A2) ) ).

% in_these_eq
tff(fact_7670_in__these__eq,axiom,
    ! [X3: product_prod_nat_nat,A2: set_op4508134149509766951at_nat] :
      ( member8440522571783428010at_nat(X3,these_8061402112538453438at_nat(A2))
    <=> member3954567711264315760at_nat(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,X3),A2) ) ).

% in_these_eq
tff(fact_7671_in__these__eq,axiom,
    ! [X3: nat,A2: set_option_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),these_nat(A2)))
    <=> member_option_nat(aa_nat_option_nat(some_nat,X3),A2) ) ).

% in_these_eq
tff(fact_7672_in__these__eq,axiom,
    ! [X3: num,A2: set_option_num] :
      ( pp(aa_set_num_bool(member_num(X3),these_num(A2)))
    <=> member_option_num(aa_num_option_num(some_num,X3),A2) ) ).

% in_these_eq
tff(fact_7673_Some__image__these__eq,axiom,
    ! [A2: set_op4508134149509766951at_nat] : image_4198897800814241419at_nat(some_P7363390416028606310at_nat,these_8061402112538453438at_nat(A2)) = collec5929226652830133682at_nat(aTP_Lamm_abw(A2)) ).

% Some_image_these_eq
tff(fact_7674_Some__image__these__eq,axiom,
    ! [A2: set_option_nat] : image_nat_option_nat(some_nat,these_nat(A2)) = collect_option_nat(aTP_Lamm_abx(A2)) ).

% Some_image_these_eq
tff(fact_7675_Some__image__these__eq,axiom,
    ! [A2: set_option_num] : image_num_option_num(some_num,these_num(A2)) = collect_option_num(aTP_Lamm_aby(A2)) ).

% Some_image_these_eq
tff(fact_7676_dual__Min,axiom,
    lattices_Min_rat(aTP_Lamm_abz) = lattic7630753665789217321ax_rat ).

% dual_Min
tff(fact_7677_dual__Min,axiom,
    lattices_Min_num(aTP_Lamm_aca) = lattic4823215512031491691ax_num ).

% dual_Min
tff(fact_7678_dual__Min,axiom,
    lattices_Min_nat(aTP_Lamm_cf) = lattic8265883725875713057ax_nat ).

% dual_Min
tff(fact_7679_dual__Min,axiom,
    lattices_Min_int(aTP_Lamm_acb) = lattic8263393255366662781ax_int ).

% dual_Min
tff(fact_7680_Un__set__drop__extend,axiom,
    ! [J: nat,L: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s3254054031482475050et_nat(L)))
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(nth_set_nat(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,zero_zero_nat)))),comple7399068483239264473et_nat(set_set_nat2(drop_set_nat(J,L)))) = comple7399068483239264473et_nat(set_set_nat2(drop_set_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,zero_zero_nat)),L))) ) ) ) ).

% Un_set_drop_extend
tff(fact_7681_subset__subseqs,axiom,
    ! [X6: set_VEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(X6),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)))
     => member_set_VEBT_VEBT(X6,image_6463372868993444447T_VEBT(set_VEBT_VEBT2,set_list_VEBT_VEBT2(subseqs_VEBT_VEBT(Xs)))) ) ).

% subset_subseqs
tff(fact_7682_subset__subseqs,axiom,
    ! [X6: set_nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,X6),aa_list_nat_set_nat(set_nat2,Xs)))
     => pp(aa_set_set_nat_bool(member_set_nat(X6),image_1775855109352712557et_nat(set_nat2,set_list_nat2(subseqs_nat(Xs))))) ) ).

% subset_subseqs
tff(fact_7683_subset__subseqs,axiom,
    ! [X6: set_real,Xs: list_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(X6),aa_lis3769085826154824284t_real(set_real2,Xs)))
     => member_set_real(X6,image_6239767680843085477t_real(set_real2,set_list_real2(subseqs_real(Xs)))) ) ).

% subset_subseqs
tff(fact_7684_subset__subseqs,axiom,
    ! [X6: set_bool,Xs: list_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(X6),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
     => member_set_bool(X6,image_9119081611755079455t_bool(set_bool2,set_list_bool2(subseqs_bool(Xs)))) ) ).

% subset_subseqs
tff(fact_7685_subset__subseqs,axiom,
    ! [X6: set_int,Xs: list_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(X6),aa_list_int_set_int(set_int2,Xs)))
     => member_set_int(X6,image_3606813740839090725et_int(set_int2,set_list_int2(subseqs_int(Xs)))) ) ).

% subset_subseqs
tff(fact_7686_image__Fpow__mono,axiom,
    ! [F: fun_nat_real,A2: set_nat,B3: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_set_nat_set_real(image_nat_real(F),A2)),B3))
     => ord_le3558479182127378552t_real(image_6333053925516494319t_real(image_nat_real(F),finite_Fpow_nat(A2)),finite_Fpow_real(B3)) ) ).

% image_Fpow_mono
tff(fact_7687_image__Fpow__mono,axiom,
    ! [F: fun_nat_set_nat,A2: set_nat,B3: set_set_nat] :
      ( pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A2)),B3))
     => ord_le9131159989063066194et_nat(image_6725021117256019401et_nat(image_nat_set_nat(F),finite_Fpow_nat(A2)),finite_Fpow_set_nat(B3)) ) ).

% image_Fpow_mono
tff(fact_7688_image__Fpow__mono,axiom,
    ! [F: fun_nat_nat,A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),A2)),B3))
     => pp(aa_set_set_nat_bool(ord_le6893508408891458716et_nat(image_7916887816326733075et_nat(image_nat_nat(F),finite_Fpow_nat(A2))),finite_Fpow_nat(B3))) ) ).

% image_Fpow_mono
tff(fact_7689_image__Fpow__mono,axiom,
    ! [F: fun_nat_int,A2: set_nat,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A2)),B3))
     => ord_le4403425263959731960et_int(image_3739036796817536367et_int(image_nat_int(F),finite_Fpow_nat(A2)),finite_Fpow_int(B3)) ) ).

% image_Fpow_mono
tff(fact_7690_image__Fpow__mono,axiom,
    ! [F: fun_int_int,A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A2)),B3))
     => ord_le4403425263959731960et_int(image_524474410958335435et_int(image_int_int(F),finite_Fpow_int(A2)),finite_Fpow_int(B3)) ) ).

% image_Fpow_mono
tff(fact_7691_set__list__bind,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VE3114949619472531401T_VEBT] : aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,bind_V4754371891896957063T_VEBT(Xs,F)) = comple2820511241208326657T_VEBT(image_2685870239581809509T_VEBT(aTP_Lamm_acc(F),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))) ).

% set_list_bind
tff(fact_7692_set__list__bind,axiom,
    ! [Xs: list_nat,F: fun_na8123536816256238275T_VEBT] : aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,bind_nat_VEBT_VEBT(Xs,F)) = comple2820511241208326657T_VEBT(image_1406951880692228733T_VEBT(aTP_Lamm_acd(F),aa_list_nat_set_nat(set_nat2,Xs))) ).

% set_list_bind
tff(fact_7693_set__list__bind,axiom,
    ! [Xs: list_real,F: fun_re8294352169103142759T_VEBT] : aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,bind_real_VEBT_VEBT(Xs,F)) = comple2820511241208326657T_VEBT(image_6925917818215209377T_VEBT(aTP_Lamm_ace(F),aa_lis3769085826154824284t_real(set_real2,Xs))) ).

% set_list_bind
tff(fact_7694_set__list__bind,axiom,
    ! [Xs: list_bool,F: fun_bo1245769998338026534T_VEBT] : aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,bind_bool_VEBT_VEBT(Xs,F)) = comple2820511241208326657T_VEBT(image_5756441063665793602T_VEBT(aTP_Lamm_acf(F),aa_lis3587183946281582166t_bool(set_bool2,Xs))) ).

% set_list_bind
tff(fact_7695_set__list__bind,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VE8613710830879766659t_real] : aa_lis3769085826154824284t_real(set_real2,bind_VEBT_VEBT_real(Xs,F)) = comple3096694443085538997t_real(image_6636839513470643793t_real(aTP_Lamm_acg(F),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))) ).

% set_list_bind
tff(fact_7696_set__list__bind,axiom,
    ! [Xs: list_nat,F: fun_nat_list_real] : aa_lis3769085826154824284t_real(set_real2,bind_nat_real(Xs,F)) = comple3096694443085538997t_real(image_nat_set_real(aTP_Lamm_ach(F),aa_list_nat_set_nat(set_nat2,Xs))) ).

% set_list_bind
tff(fact_7697_set__list__bind,axiom,
    ! [Xs: list_real,F: fun_real_list_real] : aa_lis3769085826154824284t_real(set_real2,bind_real_real(Xs,F)) = comple3096694443085538997t_real(image_real_set_real(aTP_Lamm_aci(F),aa_lis3769085826154824284t_real(set_real2,Xs))) ).

% set_list_bind
tff(fact_7698_set__list__bind,axiom,
    ! [Xs: list_bool,F: fun_bool_list_real] : aa_lis3769085826154824284t_real(set_real2,bind_bool_real(Xs,F)) = comple3096694443085538997t_real(image_bool_set_real(aTP_Lamm_acj(F),aa_lis3587183946281582166t_bool(set_bool2,Xs))) ).

% set_list_bind
tff(fact_7699_set__list__bind,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VE3624423849788874412t_bool] : aa_lis3587183946281582166t_bool(set_bool2,bind_VEBT_VEBT_bool(Xs,F)) = comple9167723193526752868t_bool(image_778176921144289224t_bool(aTP_Lamm_ack(F),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))) ).

% set_list_bind
tff(fact_7700_set__list__bind,axiom,
    ! [Xs: list_nat,F: fun_nat_list_bool] : aa_lis3587183946281582166t_bool(set_bool2,bind_nat_bool(Xs,F)) = comple9167723193526752868t_bool(image_nat_set_bool(aTP_Lamm_acl(F),aa_list_nat_set_nat(set_nat2,Xs))) ).

% set_list_bind
tff(fact_7701_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).

% pochhammer_minus'
tff(fact_7702_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).

% pochhammer_minus'
tff(fact_7703_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_7704_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_7705_power__shift,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,aa_nat_option_nat(some_nat,X3)),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,Z) ) ) ).

% power_shift
tff(fact_7706_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_7707_power__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_7708_nat__power__eq__Suc__0__iff,axiom,
    ! [X3: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X3 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_7709_nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X3))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_7710_drop0,axiom,
    ! [X5: list_bool] : drop_bool(zero_zero_nat,X5) = X5 ).

% drop0
tff(fact_7711_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_nat_real(power_power_real(A),M) = aa_nat_real(power_power_real(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_7712_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_7713_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_7714_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_7715_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).

% power_0_Suc
tff(fact_7716_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).

% power_0_Suc
tff(fact_7717_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(power_power_real(zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_7718_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_7719_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_7720_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_7721_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_7722_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_7723_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_7724_drop__Suc__Cons,axiom,
    ! [N: nat,X3: bool,Xs: list_bool] : drop_bool(aa_nat_nat(suc,N),cons_bool(X3,Xs)) = drop_bool(N,Xs) ).

% drop_Suc_Cons
tff(fact_7725_drop__Suc__Cons,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] : drop_nat(aa_nat_nat(suc,N),aa_list_nat_list_nat(cons_nat(X3),Xs)) = drop_nat(N,Xs) ).

% drop_Suc_Cons
tff(fact_7726_drop__Suc__Cons,axiom,
    ! [N: nat,X3: int,Xs: list_int] : drop_int(aa_nat_nat(suc,N),aa_list_int_list_int(cons_int(X3),Xs)) = drop_int(N,Xs) ).

% drop_Suc_Cons
tff(fact_7727_length__drop,axiom,
    ! [N: nat,Xs: list_real] : aa_list_real_nat(size_size_list_real,drop_real(N,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_real_nat(size_size_list_real,Xs)),N) ).

% length_drop
tff(fact_7728_length__drop,axiom,
    ! [N: nat,Xs: list_bool] : aa_list_bool_nat(size_size_list_bool,drop_bool(N,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_bool_nat(size_size_list_bool,Xs)),N) ).

% length_drop
tff(fact_7729_length__drop,axiom,
    ! [N: nat,Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,drop_nat(N,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_nat_nat(size_size_list_nat,Xs)),N) ).

% length_drop
tff(fact_7730_length__drop,axiom,
    ! [N: nat,Xs: list_int] : aa_list_int_nat(size_size_list_int,drop_int(N,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_int_nat(size_size_list_int,Xs)),N) ).

% length_drop
tff(fact_7731_drop__upd__irrelevant,axiom,
    ! [M: nat,N: nat,L: list_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( drop_bool(N,list_update_bool(L,M,X3)) = drop_bool(N,L) ) ) ).

% drop_upd_irrelevant
tff(fact_7732_drop__upd__irrelevant,axiom,
    ! [M: nat,N: nat,L: list_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( drop_VEBT_VEBTi(N,list_u6098035379799741383_VEBTi(L,M,X3)) = drop_VEBT_VEBTi(N,L) ) ) ).

% drop_upd_irrelevant
tff(fact_7733_drop__upd__irrelevant,axiom,
    ! [M: nat,N: nat,L: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( drop_VEBT_VEBT(N,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(L,M),X3)) = drop_VEBT_VEBT(N,L) ) ) ).

% drop_upd_irrelevant
tff(fact_7734_drop__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( drop_bool(M,list_update_bool(Xs,N,X3)) = drop_bool(M,Xs) ) ) ).

% drop_update_cancel
tff(fact_7735_drop__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( drop_VEBT_VEBTi(M,list_u6098035379799741383_VEBTi(Xs,N,X3)) = drop_VEBT_VEBTi(M,Xs) ) ) ).

% drop_update_cancel
tff(fact_7736_drop__update__cancel,axiom,
    ! [N: nat,M: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( drop_VEBT_VEBT(M,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,N),X3)) = drop_VEBT_VEBT(M,Xs) ) ) ).

% drop_update_cancel
tff(fact_7737_power__strict__increasing__iff,axiom,
    ! [B: real,X3: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B),X3)),aa_nat_real(power_power_real(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_7738_power__strict__increasing__iff,axiom,
    ! [B: rat,X3: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),X3)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_7739_power__strict__increasing__iff,axiom,
    ! [B: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_7740_power__strict__increasing__iff,axiom,
    ! [B: int,X3: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),X3)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_7741_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_7742_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_7743_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_7744_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_7745_minus__one__mult__self,axiom,
    ! [N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)) = one_one_complex ).

% minus_one_mult_self
tff(fact_7746_minus__one__mult__self,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)) = one_one_real ).

% minus_one_mult_self
tff(fact_7747_minus__one__mult__self,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)) = one_one_rat ).

% minus_one_mult_self
tff(fact_7748_minus__one__mult__self,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).

% minus_one_mult_self
tff(fact_7749_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_7750_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_7751_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(power_power_real(A),N) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_7752_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(A),N) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_7753_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( aa_nat_int(power_power_int(A),N) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_7754_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X3)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7755_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X3)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7756_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X3)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7757_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X3)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7758_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X3)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7759_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7760_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7761_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7762_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X3)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7763_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X3)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7764_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X3)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7765_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X3)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7766_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7767_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X3)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7768_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7769_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7770_last__drop,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
     => ( last_real(drop_real(N,Xs)) = last_real(Xs) ) ) ).

% last_drop
tff(fact_7771_last__drop,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( pp(aa_list_bool_bool(last_bool,drop_bool(N,Xs)))
      <=> pp(aa_list_bool_bool(last_bool,Xs)) ) ) ).

% last_drop
tff(fact_7772_last__drop,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( aa_list_nat_nat(last_nat,drop_nat(N,Xs)) = aa_list_nat_nat(last_nat,Xs) ) ) ).

% last_drop
tff(fact_7773_last__drop,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
     => ( last_int(drop_int(N,Xs)) = last_int(Xs) ) ) ).

% last_drop
tff(fact_7774_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B),M)),aa_nat_real(power_power_real(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_7775_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_7776_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_7777_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_7778_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_7779_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_7780_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_7781_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_7782_power__increasing__iff,axiom,
    ! [B: real,X3: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),X3)),aa_nat_real(power_power_real(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ) ).

% power_increasing_iff
tff(fact_7783_power__increasing__iff,axiom,
    ! [B: rat,X3: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X3)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ) ).

% power_increasing_iff
tff(fact_7784_power__increasing__iff,axiom,
    ! [B: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ) ).

% power_increasing_iff
tff(fact_7785_power__increasing__iff,axiom,
    ! [B: int,X3: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),X3)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ) ).

% power_increasing_iff
tff(fact_7786_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),N)))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7787_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N)))
    <=> ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7788_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N)))
    <=> ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7789_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N)))
    <=> ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7790_nth__drop,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(drop_VEBT_VEBTi(N,Xs)),I) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),I)) ) ) ).

% nth_drop
tff(fact_7791_nth__drop,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(drop_VEBT_VEBT(N,Xs)),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),I)) ) ) ).

% nth_drop
tff(fact_7792_nth__drop,axiom,
    ! [N: nat,Xs: list_real,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
     => ( aa_nat_real(nth_real(drop_real(N,Xs)),I) = aa_nat_real(nth_real(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),I)) ) ) ).

% nth_drop
tff(fact_7793_nth__drop,axiom,
    ! [N: nat,Xs: list_bool,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( pp(aa_nat_bool(nth_bool(drop_bool(N,Xs)),I))
      <=> pp(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),I))) ) ) ).

% nth_drop
tff(fact_7794_nth__drop,axiom,
    ! [N: nat,Xs: list_nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( aa_nat_nat(nth_nat(drop_nat(N,Xs)),I) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),I)) ) ) ).

% nth_drop
tff(fact_7795_nth__drop,axiom,
    ! [N: nat,Xs: list_int,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
     => ( aa_nat_int(nth_int(drop_int(N,Xs)),I) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),I)) ) ) ).

% nth_drop
tff(fact_7796_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),M)),aa_nat_real(power_power_real(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_7797_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_7798_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_7799_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_7800_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X3)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X3))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7801_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,X3)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X3))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7802_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X3)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X3))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7803_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X3)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X3))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7804_drop__0,axiom,
    ! [Xs: list_bool] : drop_bool(zero_zero_nat,Xs) = Xs ).

% drop_0
tff(fact_7805_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_7806_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_7807_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(power_power_real(A),N) != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_7808_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),N) != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_7809_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_7810_power__commutes,axiom,
    ! [A: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),N)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N)) ).

% power_commutes
tff(fact_7811_power__commutes,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)) ).

% power_commutes
tff(fact_7812_power__commutes,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_commutes
tff(fact_7813_power__commutes,axiom,
    ! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_commutes
tff(fact_7814_power__commutes,axiom,
    ! [A: assn,N: nat] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),A) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N)) ).

% power_commutes
tff(fact_7815_power__commutes,axiom,
    ! [A: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(A),N)),A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_nat_complex(power_power_complex(A),N)) ).

% power_commutes
tff(fact_7816_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N)) ).

% power_mult_distrib
tff(fact_7817_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ).

% power_mult_distrib
tff(fact_7818_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ).

% power_mult_distrib
tff(fact_7819_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ).

% power_mult_distrib
tff(fact_7820_power__mult__distrib,axiom,
    ! [A: assn,B: assn,N: nat] : aa_nat_assn(power_power_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),N) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),aa_nat_assn(power_power_assn(B),N)) ).

% power_mult_distrib
tff(fact_7821_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),N) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(A),N)),aa_nat_complex(power_power_complex(B),N)) ).

% power_mult_distrib
tff(fact_7822_power__commuting__commutes,axiom,
    ! [X3: real,Y: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X3) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X3),N)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(power_power_real(X3),N)) ) ) ).

% power_commuting_commutes
tff(fact_7823_power__commuting__commutes,axiom,
    ! [X3: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X3) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X3),N)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(X3),N)) ) ) ).

% power_commuting_commutes
tff(fact_7824_power__commuting__commutes,axiom,
    ! [X3: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X3) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),N)) ) ) ).

% power_commuting_commutes
tff(fact_7825_power__commuting__commutes,axiom,
    ! [X3: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),X3) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X3),N)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(X3),N)) ) ) ).

% power_commuting_commutes
tff(fact_7826_power__commuting__commutes,axiom,
    ! [X3: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X3),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),X3) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X3),N)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),aa_nat_assn(power_power_assn(X3),N)) ) ) ).

% power_commuting_commutes
tff(fact_7827_power__commuting__commutes,axiom,
    ! [X3: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),X3) )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(X3),N)),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),aa_nat_complex(power_power_complex(X3),N)) ) ) ).

% power_commuting_commutes
tff(fact_7828_power__mult,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),N) ).

% power_mult
tff(fact_7829_power__mult,axiom,
    ! [A: real,M: nat,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),M)),N) ).

% power_mult
tff(fact_7830_power__mult,axiom,
    ! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),M)),N) ).

% power_mult
tff(fact_7831_power__mult,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),M)),N) ).

% power_mult
tff(fact_7832_drop__eq__ConsD,axiom,
    ! [N: nat,Xs: list_bool,X3: bool,Xs4: list_bool] :
      ( ( drop_bool(N,Xs) = cons_bool(X3,Xs4) )
     => ( drop_bool(aa_nat_nat(suc,N),Xs) = Xs4 ) ) ).

% drop_eq_ConsD
tff(fact_7833_drop__eq__ConsD,axiom,
    ! [N: nat,Xs: list_nat,X3: nat,Xs4: list_nat] :
      ( ( drop_nat(N,Xs) = aa_list_nat_list_nat(cons_nat(X3),Xs4) )
     => ( drop_nat(aa_nat_nat(suc,N),Xs) = Xs4 ) ) ).

% drop_eq_ConsD
tff(fact_7834_drop__eq__ConsD,axiom,
    ! [N: nat,Xs: list_int,X3: int,Xs4: list_int] :
      ( ( drop_int(N,Xs) = aa_list_int_list_int(cons_int(X3),Xs4) )
     => ( drop_int(aa_nat_nat(suc,N),Xs) = Xs4 ) ) ).

% drop_eq_ConsD
tff(fact_7835_set__drop__subset,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,drop_VEBT_VEBT(N,Xs))),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))) ).

% set_drop_subset
tff(fact_7836_set__drop__subset,axiom,
    ! [N: nat,Xs: list_nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,drop_nat(N,Xs))),aa_list_nat_set_nat(set_nat2,Xs))) ).

% set_drop_subset
tff(fact_7837_set__drop__subset,axiom,
    ! [N: nat,Xs: list_real] : pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,drop_real(N,Xs))),aa_lis3769085826154824284t_real(set_real2,Xs))) ).

% set_drop_subset
tff(fact_7838_set__drop__subset,axiom,
    ! [N: nat,Xs: list_bool] : pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,drop_bool(N,Xs))),aa_lis3587183946281582166t_bool(set_bool2,Xs))) ).

% set_drop_subset
tff(fact_7839_set__drop__subset,axiom,
    ! [N: nat,Xs: list_int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,drop_int(N,Xs))),aa_list_int_set_int(set_int2,Xs))) ).

% set_drop_subset
tff(fact_7840_sorted__drop,axiom,
    ! [Xs: list_bool,N: nat] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
     => pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),drop_bool(N,Xs))) ) ).

% sorted_drop
tff(fact_7841_sorted__drop,axiom,
    ! [Xs: list_rat,N: nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => sorted_wrt_rat(ord_less_eq_rat,drop_rat(N,Xs)) ) ).

% sorted_drop
tff(fact_7842_sorted__drop,axiom,
    ! [Xs: list_num,N: nat] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => sorted_wrt_num(ord_less_eq_num,drop_num(N,Xs)) ) ).

% sorted_drop
tff(fact_7843_sorted__drop,axiom,
    ! [Xs: list_nat,N: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),drop_nat(N,Xs))) ) ).

% sorted_drop
tff(fact_7844_sorted__drop,axiom,
    ! [Xs: list_int,N: nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),drop_int(N,Xs))) ) ).

% sorted_drop
tff(fact_7845_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N))) ) ) ).

% power_mono
tff(fact_7846_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono
tff(fact_7847_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ).

% power_mono
tff(fact_7848_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono
tff(fact_7849_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N))) ) ).

% zero_le_power
tff(fact_7850_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_le_power
tff(fact_7851_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_le_power
tff(fact_7852_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_power
tff(fact_7853_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),N))) ) ).

% zero_less_power
tff(fact_7854_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_less_power
tff(fact_7855_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_less_power
tff(fact_7856_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_less_power
tff(fact_7857_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(A),N))) ) ).

% one_le_power
tff(fact_7858_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% one_le_power
tff(fact_7859_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% one_le_power
tff(fact_7860_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ).

% one_le_power
tff(fact_7861_empty__in__Fpow,axiom,
    ! [A2: set_real] : member_set_real(bot_bot_set_real,finite_Fpow_real(A2)) ).

% empty_in_Fpow
tff(fact_7862_empty__in__Fpow,axiom,
    ! [A2: set_bool] : member_set_bool(bot_bot_set_bool,finite_Fpow_bool(A2)) ).

% empty_in_Fpow
tff(fact_7863_empty__in__Fpow,axiom,
    ! [A2: set_nat] : pp(aa_set_set_nat_bool(member_set_nat(bot_bot_set_nat),finite_Fpow_nat(A2))) ).

% empty_in_Fpow
tff(fact_7864_empty__in__Fpow,axiom,
    ! [A2: set_int] : member_set_int(bot_bot_set_int,finite_Fpow_int(A2)) ).

% empty_in_Fpow
tff(fact_7865_left__right__inverse__power,axiom,
    ! [X3: real,Y: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X3),N)),aa_nat_real(power_power_real(Y),N)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_7866_left__right__inverse__power,axiom,
    ! [X3: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X3),Y) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X3),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_7867_left__right__inverse__power,axiom,
    ! [X3: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X3),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_7868_left__right__inverse__power,axiom,
    ! [X3: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X3),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_7869_left__right__inverse__power,axiom,
    ! [X3: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X3),Y) = one_one_assn )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X3),N)),aa_nat_assn(power_power_assn(Y),N)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_7870_left__right__inverse__power,axiom,
    ! [X3: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,X3),Y) = one_one_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(X3),N)),aa_nat_complex(power_power_complex(Y),N)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_7871_power__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N)) ).

% power_Suc
tff(fact_7872_power__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)) ).

% power_Suc
tff(fact_7873_power__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_Suc
tff(fact_7874_power__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_Suc
tff(fact_7875_power__Suc,axiom,
    ! [A: assn,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N)) ).

% power_Suc
tff(fact_7876_power__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_nat_complex(power_power_complex(A),N)) ).

% power_Suc
tff(fact_7877_power__Suc2,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),N)),A) ).

% power_Suc2
tff(fact_7878_power__Suc2,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),N)),A) ).

% power_Suc2
tff(fact_7879_power__Suc2,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) ).

% power_Suc2
tff(fact_7880_power__Suc2,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) ).

% power_Suc2
tff(fact_7881_power__Suc2,axiom,
    ! [A: assn,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),A) ).

% power_Suc2
tff(fact_7882_power__Suc2,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(A),N)),A) ).

% power_Suc2
tff(fact_7883_power__0,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),zero_zero_nat) = one_one_assn ).

% power_0
tff(fact_7884_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_7885_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_7886_power__0,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_7887_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_7888_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_7889_power__add,axiom,
    ! [A: real,M: nat,N: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)) ).

% power_add
tff(fact_7890_power__add,axiom,
    ! [A: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ).

% power_add
tff(fact_7891_power__add,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_7892_power__add,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ).

% power_add
tff(fact_7893_power__add,axiom,
    ! [A: assn,M: nat,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),M)),aa_nat_assn(power_power_assn(A),N)) ).

% power_add
tff(fact_7894_power__add,axiom,
    ! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ).

% power_add
tff(fact_7895_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_7896_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,drop_VEBT_VEBT(M,Xs))),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,drop_VEBT_VEBT(N,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_7897_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,drop_nat(M,Xs))),aa_list_nat_set_nat(set_nat2,drop_nat(N,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_7898_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,drop_real(M,Xs))),aa_lis3769085826154824284t_real(set_real2,drop_real(N,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_7899_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,drop_bool(M,Xs))),aa_lis3587183946281582166t_bool(set_bool2,drop_bool(N,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_7900_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,drop_int(M,Xs))),aa_list_int_set_int(set_int2,drop_int(N,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_7901_drop__update__swap,axiom,
    ! [M: nat,N: nat,Xs: list_bool,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( drop_bool(M,list_update_bool(Xs,N,X3)) = list_update_bool(drop_bool(M,Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M),X3) ) ) ).

% drop_update_swap
tff(fact_7902_drop__update__swap,axiom,
    ! [M: nat,N: nat,Xs: list_VEBT_VEBTi,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( drop_VEBT_VEBTi(M,list_u6098035379799741383_VEBTi(Xs,N,X3)) = list_u6098035379799741383_VEBTi(drop_VEBT_VEBTi(M,Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M),X3) ) ) ).

% drop_update_swap
tff(fact_7903_drop__update__swap,axiom,
    ! [M: nat,N: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( drop_VEBT_VEBT(M,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,N),X3)) = aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(drop_VEBT_VEBT(M,Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),X3) ) ) ).

% drop_update_swap
tff(fact_7904_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_7905_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_7906_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_7907_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_7908_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_7909_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),one_one_rat)) ) ) ).

% power_le_one
tff(fact_7910_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_7911_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int)) ) ) ).

% power_le_one
tff(fact_7912_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)) = aa_nat_real(power_power_real(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_7913_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_7914_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_7915_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_7916_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N))),aa_nat_real(power_power_real(B),aa_nat_nat(suc,N))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_7917_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_7918_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_7919_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,N))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_7920_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N)))) ) ).

% power_less_power_Suc
tff(fact_7921_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_less_power_Suc
tff(fact_7922_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_less_power_Suc
tff(fact_7923_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_less_power_Suc
tff(fact_7924_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N)))) ) ).

% power_gt1_lemma
tff(fact_7925_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_gt1_lemma
tff(fact_7926_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_gt1_lemma
tff(fact_7927_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_gt1_lemma
tff(fact_7928_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_7929_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_7930_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_7931_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_7932_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% power_0_left
tff(fact_7933_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% power_0_left
tff(fact_7934_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_7935_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_7936_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_7937_power__strict__increasing,axiom,
    ! [N: nat,N7: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(A),N7))) ) ) ).

% power_strict_increasing
tff(fact_7938_power__strict__increasing,axiom,
    ! [N: nat,N7: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N7))) ) ) ).

% power_strict_increasing
tff(fact_7939_power__strict__increasing,axiom,
    ! [N: nat,N7: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N7))) ) ) ).

% power_strict_increasing
tff(fact_7940_power__strict__increasing,axiom,
    ! [N: nat,N7: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N7))) ) ) ).

% power_strict_increasing
tff(fact_7941_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_7942_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_7943_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_7944_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_7945_power__minus,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(A),N)) ).

% power_minus
tff(fact_7946_power__minus,axiom,
    ! [A: real,N: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(power_power_real(A),N)) ).

% power_minus
tff(fact_7947_power__minus,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).

% power_minus
tff(fact_7948_power__minus,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(A),N)) ).

% power_minus
tff(fact_7949_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).

% zero_power
tff(fact_7950_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).

% zero_power
tff(fact_7951_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(power_power_real(zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_7952_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_7953_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_7954_power__increasing,axiom,
    ! [N: nat,N7: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(A),N7))) ) ) ).

% power_increasing
tff(fact_7955_power__increasing,axiom,
    ! [N: nat,N7: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N7))) ) ) ).

% power_increasing
tff(fact_7956_power__increasing,axiom,
    ! [N: nat,N7: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N7))) ) ) ).

% power_increasing
tff(fact_7957_power__increasing,axiom,
    ! [N: nat,N7: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N7))) ) ) ).

% power_increasing
tff(fact_7958_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),N))) ).

% zero_le_power_abs
tff(fact_7959_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N))) ).

% zero_le_power_abs
tff(fact_7960_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N))) ).

% zero_le_power_abs
tff(fact_7961_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N))) ).

% zero_le_power_abs
tff(fact_7962_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ).

% power_gt_expt
tff(fact_7963_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))) ) ).

% nat_one_le_power
tff(fact_7964_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ? [R3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
          & ( aa_nat_real(power_power_real(R3),aa_nat_nat(suc,N)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_7965_real__arch__pow__inv,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),one_one_real))
       => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X3),N2)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_7966_drop__Cons_H,axiom,
    ! [N: nat,X3: bool,Xs: list_bool] :
      ( ( ( N = zero_zero_nat )
       => ( drop_bool(N,cons_bool(X3,Xs)) = cons_bool(X3,Xs) ) )
      & ( ( N != zero_zero_nat )
       => ( drop_bool(N,cons_bool(X3,Xs)) = drop_bool(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),Xs) ) ) ) ).

% drop_Cons'
tff(fact_7967_drop__Cons_H,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( drop_nat(N,aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_list_nat_list_nat(cons_nat(X3),Xs) ) )
      & ( ( N != zero_zero_nat )
       => ( drop_nat(N,aa_list_nat_list_nat(cons_nat(X3),Xs)) = drop_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),Xs) ) ) ) ).

% drop_Cons'
tff(fact_7968_drop__Cons_H,axiom,
    ! [N: nat,X3: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( drop_int(N,aa_list_int_list_int(cons_int(X3),Xs)) = aa_list_int_list_int(cons_int(X3),Xs) ) )
      & ( ( N != zero_zero_nat )
       => ( drop_int(N,aa_list_int_list_int(cons_int(X3),Xs)) = drop_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),Xs) ) ) ) ).

% drop_Cons'
tff(fact_7969_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N))),aa_nat_real(power_power_real(A),N))) ) ) ).

% power_Suc_less
tff(fact_7970_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% power_Suc_less
tff(fact_7971_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% power_Suc_less
tff(fact_7972_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N))) ) ) ).

% power_Suc_less
tff(fact_7973_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_7974_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_7975_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_7976_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_7977_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_7978_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),one_one_rat)) ) ) ).

% power_Suc_less_one
tff(fact_7979_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_7980_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_7981_length__n__lists,axiom,
    ! [N: nat,Xs: list_real] : size_s6660260683639930848t_real(n_lists_real(N,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_list_real_nat(size_size_list_real,Xs)),N) ).

% length_n_lists
tff(fact_7982_length__n__lists,axiom,
    ! [N: nat,Xs: list_bool] : size_s9193230429364170767t_bool(n_lists_bool(N,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_list_bool_nat(size_size_list_bool,Xs)),N) ).

% length_n_lists
tff(fact_7983_length__n__lists,axiom,
    ! [N: nat,Xs: list_nat] : size_s3023201423986296836st_nat(n_lists_nat(N,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_list_nat_nat(size_size_list_nat,Xs)),N) ).

% length_n_lists
tff(fact_7984_length__n__lists,axiom,
    ! [N: nat,Xs: list_int] : size_s533118279054570080st_int(n_lists_int(N,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_list_int_nat(size_size_list_int,Xs)),N) ).

% length_n_lists
tff(fact_7985_power__strict__decreasing,axiom,
    ! [N: nat,N7: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N7)),aa_nat_real(power_power_real(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_7986_power__strict__decreasing,axiom,
    ! [N: nat,N7: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N7)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_7987_power__strict__decreasing,axiom,
    ! [N: nat,N7: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N7)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_7988_power__strict__decreasing,axiom,
    ! [N: nat,N7: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N7))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N7)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_7989_power__decreasing,axiom,
    ! [N: nat,N7: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N7)),aa_nat_real(power_power_real(A),N))) ) ) ) ).

% power_decreasing
tff(fact_7990_power__decreasing,axiom,
    ! [N: nat,N7: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N7)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_decreasing
tff(fact_7991_power__decreasing,axiom,
    ! [N: nat,N7: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N7)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_decreasing
tff(fact_7992_power__decreasing,axiom,
    ! [N: nat,N7: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N7))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N7)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_decreasing
tff(fact_7993_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( ( aa_nat_real(power_power_real(A),N) = aa_nat_real(power_power_real(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_7994_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_7995_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_7996_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_7997_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),N) = aa_nat_real(power_power_real(B),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_7998_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_7999_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_8000_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_8001_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_8002_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_8003_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_8004_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_8005_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(power_power_real(A),N))) ) ) ).

% self_le_power
tff(fact_8006_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% self_le_power
tff(fact_8007_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% self_le_power
tff(fact_8008_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N))) ) ) ).

% self_le_power
tff(fact_8009_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),N))) ) ) ).

% one_less_power
tff(fact_8010_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% one_less_power
tff(fact_8011_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% one_less_power
tff(fact_8012_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ) ).

% one_less_power
tff(fact_8013_Fpow__mono,axiom,
    ! [A2: set_int,B3: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B3))
     => ord_le4403425263959731960et_int(finite_Fpow_int(A2),finite_Fpow_int(B3)) ) ).

% Fpow_mono
tff(fact_8014_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ( aa_nat_real(power_power_real(R3),N) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_8015_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
            & ( aa_nat_real(power_power_real(X),N) = A )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4))
                  & ( aa_nat_real(power_power_real(Y4),N) = A ) )
               => ( Y4 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_8016_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))) ) ) ).

% nat_mult_power_less_eq
tff(fact_8017_Fpow__def,axiom,
    ! [A2: set_nat] : finite_Fpow_nat(A2) = collect_set_nat(aa_set5116768664143533222t_bool(aTP_Lamm_acm,A2)) ).

% Fpow_def
tff(fact_8018_Fpow__def,axiom,
    ! [A2: set_complex] : finite_Fpow_complex(A2) = collect_set_complex(aTP_Lamm_acn(A2)) ).

% Fpow_def
tff(fact_8019_Fpow__def,axiom,
    ! [A2: set_Code_integer] : finite1532502677820914807nteger(A2) = collec574505750873337192nteger(aTP_Lamm_aco(A2)) ).

% Fpow_def
tff(fact_8020_Fpow__def,axiom,
    ! [A2: set_int] : finite_Fpow_int(A2) = collect_set_int(aTP_Lamm_acp(A2)) ).

% Fpow_def
tff(fact_8021_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( cons_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I),drop_VEBT_VEBTi(aa_nat_nat(suc,I),Xs)) = drop_VEBT_VEBTi(I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_8022_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( cons_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I),drop_VEBT_VEBT(aa_nat_nat(suc,I),Xs)) = drop_VEBT_VEBT(I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_8023_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
     => ( cons_real(aa_nat_real(nth_real(Xs),I),drop_real(aa_nat_nat(suc,I),Xs)) = drop_real(I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_8024_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( cons_bool(aa_nat_bool(nth_bool(Xs),I),drop_bool(aa_nat_nat(suc,I),Xs)) = drop_bool(I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_8025_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( aa_list_nat_list_nat(cons_nat(aa_nat_nat(nth_nat(Xs),I)),drop_nat(aa_nat_nat(suc,I),Xs)) = drop_nat(I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_8026_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( aa_list_int_list_int(cons_int(aa_nat_int(nth_int(Xs),I)),drop_int(aa_nat_nat(suc,I),Xs)) = drop_int(I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_8027_in__set__drop__conv__nth,axiom,
    ! [X3: vEBT_VEBTi,N: nat,L: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(drop_VEBT_VEBTi(N,L)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I3))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(L)))
          & ( X3 = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I3) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_8028_in__set__drop__conv__nth,axiom,
    ! [X3: set_nat,N: nat,L: list_set_nat] :
      ( pp(aa_set_set_nat_bool(member_set_nat(X3),set_set_nat2(drop_set_nat(N,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I3))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3254054031482475050et_nat(L)))
          & ( X3 = aa_nat_set_nat(nth_set_nat(L),I3) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_8029_in__set__drop__conv__nth,axiom,
    ! [X3: vEBT_VEBT,N: nat,L: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,drop_VEBT_VEBT(N,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I3))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(L)))
          & ( X3 = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I3) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_8030_in__set__drop__conv__nth,axiom,
    ! [X3: real,N: nat,L: list_real] :
      ( pp(aa_set_real_bool(member_real(X3),aa_lis3769085826154824284t_real(set_real2,drop_real(N,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I3))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_real_nat(size_size_list_real,L)))
          & ( X3 = aa_nat_real(nth_real(L),I3) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_8031_in__set__drop__conv__nth,axiom,
    ! [X3: bool,N: nat,L: list_bool] :
      ( pp(aa_set_bool_bool(member_bool(X3),aa_lis3587183946281582166t_bool(set_bool2,drop_bool(N,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I3))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_bool_nat(size_size_list_bool,L)))
          & ( pp(X3)
          <=> pp(aa_nat_bool(nth_bool(L),I3)) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_8032_in__set__drop__conv__nth,axiom,
    ! [X3: nat,N: nat,L: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X3),aa_list_nat_set_nat(set_nat2,drop_nat(N,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I3))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,L)))
          & ( X3 = aa_nat_nat(nth_nat(L),I3) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_8033_in__set__drop__conv__nth,axiom,
    ! [X3: int,N: nat,L: list_int] :
      ( pp(aa_set_int_bool(member_int(X3),aa_list_int_set_int(set_int2,drop_int(N,L))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I3))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_list_int_nat(size_size_list_int,L)))
          & ( X3 = aa_nat_int(nth_int(L),I3) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_8034_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N)),aa_nat_real(power_power_real(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_8035_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_8036_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_8037_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_8038_power__eq__if,axiom,
    ! [M: nat,P3: real] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_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_8039_power__eq__if,axiom,
    ! [M: nat,P3: rat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_8040_power__eq__if,axiom,
    ! [M: nat,P3: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_8041_power__eq__if,axiom,
    ! [M: nat,P3: int] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_8042_power__eq__if,axiom,
    ! [M: nat,P3: assn] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_assn(power_power_assn(P3),M) = one_one_assn ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_assn(power_power_assn(P3),M) = 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_8043_power__eq__if,axiom,
    ! [M: nat,P3: complex] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = aa_complex_complex(aa_com3342990543769683525omplex(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_8044_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_real(power_power_real(A),N) ) ) ).

% power_minus_mult
tff(fact_8045_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_minus_mult
tff(fact_8046_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_8047_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_minus_mult
tff(fact_8048_power__minus__mult,axiom,
    ! [N: nat,A: assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( 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,N),one_one_nat))),A) = aa_nat_assn(power_power_assn(A),N) ) ) ).

% power_minus_mult
tff(fact_8049_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% power_minus_mult
tff(fact_8050_sgn__power__injE,axiom,
    ! [A: real,N: nat,X3: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N)) = X3 )
     => ( ( X3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,B)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_8051_linear__plus__1__le__power,axiom,
    ! [X3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X3)),one_one_real)),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),one_one_real)),N))) ) ).

% linear_plus_1_le_power
tff(fact_8052_Bernoulli__inequality,axiom,
    ! [X3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X3))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X3)),N))) ) ).

% Bernoulli_inequality
tff(fact_8053_pochhammer__minus,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_8054_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_8055_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_8056_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_8057_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_8058_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_8059_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_8060_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_8061_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_acq(N),C)))) ) ).

% finite_nth_roots
tff(fact_8062_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_acr,N)))) ) ).

% finite_roots_unity
tff(fact_8063_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_acs,N)))) ) ).

% finite_roots_unity
tff(fact_8064_drop__last__conv,axiom,
    ! [L: list_real] :
      ( ( L != nil_real )
     => ( drop_real(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_real_nat(size_size_list_real,L)),aa_nat_nat(suc,zero_zero_nat)),L) = cons_real(last_real(L),nil_real) ) ) ).

% drop_last_conv
tff(fact_8065_drop__last__conv,axiom,
    ! [L: list_bool] :
      ( ( L != nil_bool )
     => ( drop_bool(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_bool_nat(size_size_list_bool,L)),aa_nat_nat(suc,zero_zero_nat)),L) = cons_bool(aa_list_bool_bool(last_bool,L),nil_bool) ) ) ).

% drop_last_conv
tff(fact_8066_drop__last__conv,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
     => ( drop_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) = aa_list_nat_list_nat(cons_nat(aa_list_nat_nat(last_nat,L)),nil_nat) ) ) ).

% drop_last_conv
tff(fact_8067_drop__last__conv,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
     => ( drop_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_int_nat(size_size_list_int,L)),aa_nat_nat(suc,zero_zero_nat)),L) = aa_list_int_list_int(cons_int(last_int(L)),nil_int) ) ) ).

% drop_last_conv
tff(fact_8068_horner__sum__eq__sum,axiom,
    ! [F: fun_VEBT_VEBTi_rat,A: rat,Xs: list_VEBT_VEBTi] : groups4473738407398356458Ti_rat(F,A,Xs) = groups2906978787729119204at_rat(aa_lis5007409733734533800at_rat(aa_rat8420674886958173181at_rat(aTP_Lamm_act(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))) ).

% horner_sum_eq_sum
tff(fact_8069_horner__sum__eq__sum,axiom,
    ! [F: fun_VEBT_VEBT_rat,A: rat,Xs: list_VEBT_VEBT] : groups6210023459537951729BT_rat(F,A,Xs) = groups2906978787729119204at_rat(aa_lis4303274161602456867at_rat(aa_rat7802996917357006812at_rat(aTP_Lamm_acu(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum
tff(fact_8070_horner__sum__eq__sum,axiom,
    ! [F: fun_real_rat,A: rat,Xs: list_real] : groups7414964958337966817al_rat(F,A,Xs) = groups2906978787729119204at_rat(aa_lis2767325350024641823at_rat(aa_rat1858428167516265844at_rat(aTP_Lamm_acv(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs))) ).

% horner_sum_eq_sum
tff(fact_8071_horner__sum__eq__sum,axiom,
    ! [F: fun_bool_rat,A: rat,Xs: list_bool] : groups3416007005163685332ol_rat(F,A,Xs) = groups2906978787729119204at_rat(aa_lis1392928297027365766at_rat(aa_rat954805563275384383at_rat(aTP_Lamm_acw(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_bool_nat(size_size_list_bool,Xs))) ).

% horner_sum_eq_sum
tff(fact_8072_horner__sum__eq__sum,axiom,
    ! [F: fun_nat_rat,A: rat,Xs: list_nat] : groups6853238114764508677at_rat(F,A,Xs) = groups2906978787729119204at_rat(aa_lis7014846069874721091at_rat(aa_rat4396832037704353944at_rat(aTP_Lamm_acx(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% horner_sum_eq_sum
tff(fact_8073_horner__sum__eq__sum,axiom,
    ! [F: fun_int_rat,A: rat,Xs: list_int] : groups7852591826665563233nt_rat(F,A,Xs) = groups2906978787729119204at_rat(aa_lis1742918661435654559at_rat(aa_rat8348276666120063220at_rat(aTP_Lamm_acy(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_int_nat(size_size_list_int,Xs))) ).

% horner_sum_eq_sum
tff(fact_8074_horner__sum__eq__sum,axiom,
    ! [F: fun_VEBT_VEBTi_int,A: int,Xs: list_VEBT_VEBTi] : groups5106377996975801918Ti_int(F,A,Xs) = groups3539618377306564664at_int(aa_lis4337787272618733820at_int(aa_int7846649280651917989at_int(aTP_Lamm_acz(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs))) ).

% horner_sum_eq_sum
tff(fact_8075_horner__sum__eq__sum,axiom,
    ! [F: fun_VEBT_VEBT_int,A: int,Xs: list_VEBT_VEBT] : groups6842663049115397189BT_int(F,A,Xs) = groups3539618377306564664at_int(aa_lis3633651700486656887at_int(aa_int4104624148011801308at_int(aTP_Lamm_ada(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ).

% horner_sum_eq_sum
tff(fact_8076_horner__sum__eq__sum,axiom,
    ! [F: fun_real_int,A: int,Xs: list_real] : groups8047604547915412277al_int(F,A,Xs) = groups3539618377306564664at_int(aa_lis2097702888908841843at_int(aa_int464142701096502812at_int(aTP_Lamm_adb(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs))) ).

% horner_sum_eq_sum
tff(fact_8077_horner__sum__eq__sum,axiom,
    ! [F: fun_nat_int,A: int,Xs: list_nat] : groups7485877704341954137at_int(F,A,Xs) = groups3539618377306564664at_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_adc(F),A),Xs),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ).

% horner_sum_eq_sum
tff(fact_8078_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( list_u6098035379799741383_VEBTi(Xs,I,A) = append_VEBT_VEBTi(take_VEBT_VEBTi(I,Xs),cons_VEBT_VEBTi(A,drop_VEBT_VEBTi(aa_nat_nat(suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_8079_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,A: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(Xs,I),A) = append_VEBT_VEBT(take_VEBT_VEBT(I,Xs),cons_VEBT_VEBT(A,drop_VEBT_VEBT(aa_nat_nat(suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_8080_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs: list_real,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,Xs)))
     => ( aa_real_list_real(list_update_real(Xs,I),A) = append_real(take_real(I,Xs),cons_real(A,drop_real(aa_nat_nat(suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_8081_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs: list_bool,A: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( list_update_bool(Xs,I,A) = append_bool(take_bool(I,Xs),cons_bool(A,drop_bool(aa_nat_nat(suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_8082_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs: list_nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( list_update_nat(Xs,I,A) = append_nat(take_nat(I,Xs),aa_list_nat_list_nat(cons_nat(A),drop_nat(aa_nat_nat(suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_8083_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs: list_int,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,Xs)))
     => ( list_update_int(Xs,I,A) = append_int(take_int(I,Xs),aa_list_int_list_int(cons_int(A),drop_int(aa_nat_nat(suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_8084_concat__inth,axiom,
    ! [Xs: list_VEBT_VEBTi,X3: vEBT_VEBTi,Ys: list_VEBT_VEBTi] : aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(append_VEBT_VEBTi(Xs,append_VEBT_VEBTi(cons_VEBT_VEBTi(X3,nil_VEBT_VEBTi),Ys))),size_s7982070591426661849_VEBTi(Xs)) = X3 ).

% concat_inth
tff(fact_8085_concat__inth,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT,Ys: list_VEBT_VEBT] : aa_nat_VEBT_VEBT(nth_VEBT_VEBT(append_VEBT_VEBT(Xs,append_VEBT_VEBT(cons_VEBT_VEBT(X3,nil_VEBT_VEBT),Ys))),size_s6755466524823107622T_VEBT(Xs)) = X3 ).

% concat_inth
tff(fact_8086_concat__inth,axiom,
    ! [Xs: list_real,X3: real,Ys: list_real] : aa_nat_real(nth_real(append_real(Xs,append_real(cons_real(X3,nil_real),Ys))),aa_list_real_nat(size_size_list_real,Xs)) = X3 ).

% concat_inth
tff(fact_8087_concat__inth,axiom,
    ! [Xs: list_bool,X3: bool,Ys: list_bool] :
      ( pp(aa_nat_bool(nth_bool(append_bool(Xs,append_bool(cons_bool(X3,nil_bool),Ys))),aa_list_bool_nat(size_size_list_bool,Xs)))
    <=> pp(X3) ) ).

% concat_inth
tff(fact_8088_concat__inth,axiom,
    ! [Xs: list_nat,X3: nat,Ys: list_nat] : aa_nat_nat(nth_nat(append_nat(Xs,append_nat(aa_list_nat_list_nat(cons_nat(X3),nil_nat),Ys))),aa_list_nat_nat(size_size_list_nat,Xs)) = X3 ).

% concat_inth
tff(fact_8089_concat__inth,axiom,
    ! [Xs: list_int,X3: int,Ys: list_int] : aa_nat_int(nth_int(append_int(Xs,append_int(aa_list_int_list_int(cons_int(X3),nil_int),Ys))),aa_list_int_nat(size_size_list_int,Xs)) = X3 ).

% concat_inth
tff(fact_8090_pos__n__replace,axiom,
    ! [N: nat,Xs: list_real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
     => ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,append_real(take_real(N,Xs),append_real(cons_real(Y,nil_real),drop_real(aa_nat_nat(suc,N),Xs)))) ) ) ).

% pos_n_replace
tff(fact_8091_pos__n__replace,axiom,
    ! [N: nat,Xs: list_bool,Y: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,append_bool(take_bool(N,Xs),append_bool(cons_bool(Y,nil_bool),drop_bool(aa_nat_nat(suc,N),Xs)))) ) ) ).

% pos_n_replace
tff(fact_8092_pos__n__replace,axiom,
    ! [N: nat,Xs: list_nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,append_nat(take_nat(N,Xs),append_nat(aa_list_nat_list_nat(cons_nat(Y),nil_nat),drop_nat(aa_nat_nat(suc,N),Xs)))) ) ) ).

% pos_n_replace
tff(fact_8093_pos__n__replace,axiom,
    ! [N: nat,Xs: list_int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
     => ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,append_int(take_int(N,Xs),append_int(aa_list_int_list_int(cons_int(Y),nil_int),drop_int(aa_nat_nat(suc,N),Xs)))) ) ) ).

% pos_n_replace
tff(fact_8094_append__eq__append__conv,axiom,
    ! [Xs: list_real,Ys: list_real,Us: list_real,Vs: list_real] :
      ( ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
        | ( aa_list_real_nat(size_size_list_real,Us) = aa_list_real_nat(size_size_list_real,Vs) ) )
     => ( ( append_real(Xs,Us) = append_real(Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_8095_append__eq__append__conv,axiom,
    ! [Xs: list_bool,Ys: list_bool,Us: list_bool,Vs: list_bool] :
      ( ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
        | ( aa_list_bool_nat(size_size_list_bool,Us) = aa_list_bool_nat(size_size_list_bool,Vs) ) )
     => ( ( append_bool(Xs,Us) = append_bool(Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_8096_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
        | ( aa_list_nat_nat(size_size_list_nat,Us) = aa_list_nat_nat(size_size_list_nat,Vs) ) )
     => ( ( append_nat(Xs,Us) = append_nat(Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_8097_append__eq__append__conv,axiom,
    ! [Xs: list_int,Ys: list_int,Us: list_int,Vs: list_int] :
      ( ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
        | ( aa_list_int_nat(size_size_list_int,Us) = aa_list_int_nat(size_size_list_int,Vs) ) )
     => ( ( append_int(Xs,Us) = append_int(Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_8098_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ak),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_8099_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_am,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_8100_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_add),A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_8101_sum_Oneutral__const,axiom,
    ! [A2: set_int] : aa_set_int_int(groups4538972089207619220nt_int(aTP_Lamm_al),A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_8102_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_ade(F),A2) ).

% of_nat_sum
tff(fact_8103_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = aa_set_int_int(groups4538972089207619220nt_int(aTP_Lamm_adf(F)),A2) ).

% of_nat_sum
tff(fact_8104_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_adg(F),A2) ).

% of_nat_sum
tff(fact_8105_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_adh(F)),A2) ).

% of_nat_sum
tff(fact_8106_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pv(F)),A2) ).

% of_nat_sum
tff(fact_8107_abs__sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adi(F)),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adi(F)),A2) ).

% abs_sum_abs
tff(fact_8108_abs__sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(abs_abs_int,aa_set_int_int(groups4538972089207619220nt_int(aTP_Lamm_adj(F)),A2)) = aa_set_int_int(groups4538972089207619220nt_int(aTP_Lamm_adj(F)),A2) ).

% abs_sum_abs
tff(fact_8109_nth__repl,axiom,
    ! [M: nat,Xs: list_VEBT_VEBTi,N: nat,X3: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
       => ( ( M != N )
         => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(append_VEBT_VEBTi(take_VEBT_VEBTi(N,Xs),append_VEBT_VEBTi(cons_VEBT_VEBTi(X3,nil_VEBT_VEBTi),drop_VEBT_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),Xs)))),M) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_8110_nth__repl,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N: nat,X3: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => ( ( M != N )
         => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(append_VEBT_VEBT(take_VEBT_VEBT(N,Xs),append_VEBT_VEBT(cons_VEBT_VEBT(X3,nil_VEBT_VEBT),drop_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),Xs)))),M) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_8111_nth__repl,axiom,
    ! [M: nat,Xs: list_real,N: nat,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_real_nat(size_size_list_real,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_real_nat(size_size_list_real,Xs)))
       => ( ( M != N )
         => ( aa_nat_real(nth_real(append_real(take_real(N,Xs),append_real(cons_real(X3,nil_real),drop_real(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),Xs)))),M) = aa_nat_real(nth_real(Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_8112_nth__repl,axiom,
    ! [M: nat,Xs: list_bool,N: nat,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_bool_nat(size_size_list_bool,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)))
       => ( ( M != N )
         => ( pp(aa_nat_bool(nth_bool(append_bool(take_bool(N,Xs),append_bool(cons_bool(X3,nil_bool),drop_bool(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),Xs)))),M))
          <=> pp(aa_nat_bool(nth_bool(Xs),M)) ) ) ) ) ).

% nth_repl
tff(fact_8113_nth__repl,axiom,
    ! [M: nat,Xs: list_nat,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_nat_nat(size_size_list_nat,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)))
       => ( ( M != N )
         => ( aa_nat_nat(nth_nat(append_nat(take_nat(N,Xs),append_nat(aa_list_nat_list_nat(cons_nat(X3),nil_nat),drop_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),Xs)))),M) = aa_nat_nat(nth_nat(Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_8114_nth__repl,axiom,
    ! [M: nat,Xs: list_int,N: nat,X3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_list_int_nat(size_size_list_int,Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_list_int_nat(size_size_list_int,Xs)))
       => ( ( M != N )
         => ( aa_nat_int(nth_int(append_int(take_int(N,Xs),append_int(aa_list_int_list_int(cons_int(X3),nil_int),drop_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat),Xs)))),M) = aa_nat_int(nth_int(Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_8115_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_8116_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_8117_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_8118_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_8119_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_8120_sum_Oempty,axiom,
    ! [G: fun_bool_complex] : groups2959343415743557771omplex(G,bot_bot_set_bool) = zero_zero_complex ).

% sum.empty
tff(fact_8121_sum_Oempty,axiom,
    ! [G: fun_bool_real] : groups502887708668150281l_real(G,bot_bot_set_bool) = zero_zero_real ).

% sum.empty
tff(fact_8122_sum_Oempty,axiom,
    ! [G: fun_bool_rat] : groups5133371213642249141ol_rat(G,bot_bot_set_bool) = zero_zero_rat ).

% sum.empty
tff(fact_8123_sum_Oempty,axiom,
    ! [G: fun_bool_nat] : groups5768501273728744877ol_nat(G,bot_bot_set_bool) = zero_zero_nat ).

% sum.empty
tff(fact_8124_sum_Oempty,axiom,
    ! [G: fun_bool_int] : groups5766010803219694601ol_int(G,bot_bot_set_bool) = zero_zero_int ).

% sum.empty
tff(fact_8125_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2073611262835488442omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_8126_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_8127_sum_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: fun_Co1900716531650368264omplex] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( groups8024822376189712711omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_8128_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_8129_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_8130_sum_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( groups1270011288395367621r_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_8131_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2906978787729119204at_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8132_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8133_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8134_sum_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: fun_Code_integer_rat] :
      ( ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( groups6602215022474089585er_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8135_sum__eq__0__iff,axiom,
    ! [F4: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,F4))
     => ( ( groups4541462559716669496nt_nat(F,F4) = zero_zero_nat )
      <=> ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),F4))
           => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8136_sum__eq__0__iff,axiom,
    ! [F4: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F4))
     => ( ( groups5693394587270226106ex_nat(F,F4) = zero_zero_nat )
      <=> ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),F4))
           => ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8137_sum__eq__0__iff,axiom,
    ! [F4: set_Code_integer,F: fun_Code_integer_nat] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,F4))
     => ( ( groups7237345082560585321er_nat(F,F4) = zero_zero_nat )
      <=> ! [X4: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X4),F4))
           => ( aa_Code_integer_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8138_sum__eq__0__iff,axiom,
    ! [F4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F4))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F4) = zero_zero_nat )
      <=> ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),F4))
           => ( aa_nat_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8139_length__0__conv,axiom,
    ! [Xs: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = zero_zero_nat )
    <=> ( Xs = nil_real ) ) ).

% length_0_conv
tff(fact_8140_length__0__conv,axiom,
    ! [Xs: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = zero_zero_nat )
    <=> ( Xs = nil_bool ) ) ).

% length_0_conv
tff(fact_8141_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = zero_zero_nat )
    <=> ( Xs = nil_nat ) ) ).

% length_0_conv
tff(fact_8142_length__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = zero_zero_nat )
    <=> ( Xs = nil_int ) ) ).

% length_0_conv
tff(fact_8143_set__empty2,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( bot_bo8194388402131092736T_VEBT = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs) )
    <=> ( Xs = nil_VEBT_VEBT ) ) ).

% set_empty2
tff(fact_8144_set__empty2,axiom,
    ! [Xs: list_real] :
      ( ( bot_bot_set_real = aa_lis3769085826154824284t_real(set_real2,Xs) )
    <=> ( Xs = nil_real ) ) ).

% set_empty2
tff(fact_8145_set__empty2,axiom,
    ! [Xs: list_bool] :
      ( ( bot_bot_set_bool = aa_lis3587183946281582166t_bool(set_bool2,Xs) )
    <=> ( Xs = nil_bool ) ) ).

% set_empty2
tff(fact_8146_set__empty2,axiom,
    ! [Xs: list_nat] :
      ( ( bot_bot_set_nat = aa_list_nat_set_nat(set_nat2,Xs) )
    <=> ( Xs = nil_nat ) ) ).

% set_empty2
tff(fact_8147_set__empty2,axiom,
    ! [Xs: list_int] :
      ( ( bot_bot_set_int = aa_list_int_set_int(set_int2,Xs) )
    <=> ( Xs = nil_int ) ) ).

% set_empty2
tff(fact_8148_set__empty,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs) = bot_bo8194388402131092736T_VEBT )
    <=> ( Xs = nil_VEBT_VEBT ) ) ).

% set_empty
tff(fact_8149_set__empty,axiom,
    ! [Xs: list_real] :
      ( ( aa_lis3769085826154824284t_real(set_real2,Xs) = bot_bot_set_real )
    <=> ( Xs = nil_real ) ) ).

% set_empty
tff(fact_8150_set__empty,axiom,
    ! [Xs: list_bool] :
      ( ( aa_lis3587183946281582166t_bool(set_bool2,Xs) = bot_bot_set_bool )
    <=> ( Xs = nil_bool ) ) ).

% set_empty
tff(fact_8151_set__empty,axiom,
    ! [Xs: list_nat] :
      ( ( aa_list_nat_set_nat(set_nat2,Xs) = bot_bot_set_nat )
    <=> ( Xs = nil_nat ) ) ).

% set_empty
tff(fact_8152_set__empty,axiom,
    ! [Xs: list_int] :
      ( ( aa_list_int_set_int(set_int2,Xs) = bot_bot_set_int )
    <=> ( Xs = nil_int ) ) ).

% set_empty
tff(fact_8153_length__append,axiom,
    ! [Xs: list_real,Ys: list_real] : aa_list_real_nat(size_size_list_real,append_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_real_nat(size_size_list_real,Ys)) ).

% length_append
tff(fact_8154_length__append,axiom,
    ! [Xs: list_bool,Ys: list_bool] : aa_list_bool_nat(size_size_list_bool,append_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_bool_nat(size_size_list_bool,Xs)),aa_list_bool_nat(size_size_list_bool,Ys)) ).

% length_append
tff(fact_8155_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] : aa_list_nat_nat(size_size_list_nat,append_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)) ).

% length_append
tff(fact_8156_length__append,axiom,
    ! [Xs: list_int,Ys: list_int] : aa_list_int_nat(size_size_list_int,append_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_int_nat(size_size_list_int,Xs)),aa_list_int_nat(size_size_list_int,Ys)) ).

% length_append
tff(fact_8157_take0,axiom,
    ! [X5: list_bool] : take_bool(zero_zero_nat,X5) = nil_bool ).

% take0
tff(fact_8158_take0,axiom,
    ! [X5: list_nat] : take_nat(zero_zero_nat,X5) = nil_nat ).

% take0
tff(fact_8159_take0,axiom,
    ! [X5: list_int] : take_int(zero_zero_nat,X5) = nil_int ).

% take0
tff(fact_8160_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( take_bool(N,Xs) = nil_bool )
    <=> ( ( N = zero_zero_nat )
        | ( Xs = nil_bool ) ) ) ).

% take_eq_Nil
tff(fact_8161_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( take_nat(N,Xs) = nil_nat )
    <=> ( ( N = zero_zero_nat )
        | ( Xs = nil_nat ) ) ) ).

% take_eq_Nil
tff(fact_8162_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( take_int(N,Xs) = nil_int )
    <=> ( ( N = zero_zero_nat )
        | ( Xs = nil_int ) ) ) ).

% take_eq_Nil
tff(fact_8163_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( nil_bool = take_bool(N,Xs) )
    <=> ( ( N = zero_zero_nat )
        | ( Xs = nil_bool ) ) ) ).

% take_eq_Nil2
tff(fact_8164_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( nil_nat = take_nat(N,Xs) )
    <=> ( ( N = zero_zero_nat )
        | ( Xs = nil_nat ) ) ) ).

% take_eq_Nil2
tff(fact_8165_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( nil_int = take_int(N,Xs) )
    <=> ( ( N = zero_zero_nat )
        | ( Xs = nil_int ) ) ) ).

% take_eq_Nil2
tff(fact_8166_replicate__empty,axiom,
    ! [N: nat,X3: nat] :
      ( ( replicate_nat(N,X3) = nil_nat )
    <=> ( N = zero_zero_nat ) ) ).

% replicate_empty
tff(fact_8167_replicate__empty,axiom,
    ! [N: nat,X3: int] :
      ( ( replicate_int(N,X3) = nil_int )
    <=> ( N = zero_zero_nat ) ) ).

% replicate_empty
tff(fact_8168_replicate__empty,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3) = nil_VEBT_VEBT )
    <=> ( N = zero_zero_nat ) ) ).

% replicate_empty
tff(fact_8169_replicate__empty,axiom,
    ! [N: nat,X3: bool] :
      ( ( replicate_bool(N,X3) = nil_bool )
    <=> ( N = zero_zero_nat ) ) ).

% replicate_empty
tff(fact_8170_empty__replicate,axiom,
    ! [N: nat,X3: nat] :
      ( ( nil_nat = replicate_nat(N,X3) )
    <=> ( N = zero_zero_nat ) ) ).

% empty_replicate
tff(fact_8171_empty__replicate,axiom,
    ! [N: nat,X3: int] :
      ( ( nil_int = replicate_int(N,X3) )
    <=> ( N = zero_zero_nat ) ) ).

% empty_replicate
tff(fact_8172_empty__replicate,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( nil_VEBT_VEBT = aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3) )
    <=> ( N = zero_zero_nat ) ) ).

% empty_replicate
tff(fact_8173_empty__replicate,axiom,
    ! [N: nat,X3: bool] :
      ( ( nil_bool = replicate_bool(N,X3) )
    <=> ( N = zero_zero_nat ) ) ).

% empty_replicate
tff(fact_8174_zip__append,axiom,
    ! [Xs: list_real,Us: list_real,Ys: list_real,Vs: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Us) )
     => ( zip_real_real(append_real(Xs,Ys),append_real(Us,Vs)) = append8511633455026463060l_real(zip_real_real(Xs,Us),zip_real_real(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8175_zip__append,axiom,
    ! [Xs: list_real,Us: list_bool,Ys: list_real,Vs: list_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Us) )
     => ( zip_real_bool(append_real(Xs,Ys),append_bool(Us,Vs)) = append7873561321238512645l_bool(zip_real_bool(Xs,Us),zip_real_bool(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8176_zip__append,axiom,
    ! [Xs: list_real,Us: list_nat,Ys: list_real,Vs: list_nat] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_nat_nat(size_size_list_nat,Us) )
     => ( zip_real_nat(append_real(Xs,Ys),append_nat(Us,Vs)) = append2703634050814417656al_nat(zip_real_nat(Xs,Us),zip_real_nat(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8177_zip__append,axiom,
    ! [Xs: list_real,Us: list_int,Ys: list_real,Vs: list_int] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_int_nat(size_size_list_int,Us) )
     => ( zip_real_int(append_real(Xs,Ys),append_int(Us,Vs)) = append7749155068159996756al_int(zip_real_int(Xs,Us),zip_real_int(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8178_zip__append,axiom,
    ! [Xs: list_bool,Us: list_real,Ys: list_bool,Vs: list_real] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_real_nat(size_size_list_real,Us) )
     => ( zip_bool_real(append_bool(Xs,Ys),append_real(Us,Vs)) = append8258889960777667607l_real(zip_bool_real(Xs,Us),zip_bool_real(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8179_zip__append,axiom,
    ! [Xs: list_bool,Us: list_bool,Ys: list_bool,Vs: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Us) )
     => ( zip_bool_bool(append_bool(Xs,Ys),append_bool(Us,Vs)) = append1767198414729310338l_bool(zip_bool_bool(Xs,Us),zip_bool_bool(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8180_zip__append,axiom,
    ! [Xs: list_bool,Us: list_nat,Ys: list_bool,Vs: list_nat] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_nat_nat(size_size_list_nat,Us) )
     => ( zip_bool_nat(append_bool(Xs,Ys),append_nat(Us,Vs)) = append4937290130733399867ol_nat(zip_bool_nat(Xs,Us),zip_bool_nat(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8181_zip__append,axiom,
    ! [Xs: list_bool,Us: list_int,Ys: list_bool,Vs: list_int] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_int_nat(size_size_list_int,Us) )
     => ( zip_bool_int(append_bool(Xs,Ys),append_int(Us,Vs)) = append759439111224203159ol_int(zip_bool_int(Xs,Us),zip_bool_int(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8182_zip__append,axiom,
    ! [Xs: list_nat,Us: list_real,Ys: list_nat,Vs: list_real] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_real_nat(size_size_list_real,Us) )
     => ( zip_nat_real(append_nat(Xs,Ys),append_real(Us,Vs)) = append6678681742291297912t_real(zip_nat_real(Xs,Us),zip_nat_real(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8183_zip__append,axiom,
    ! [Xs: list_nat,Us: list_bool,Ys: list_nat,Vs: list_bool] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_bool_nat(size_size_list_bool,Us) )
     => ( zip_nat_bool(append_nat(Xs,Ys),append_bool(Us,Vs)) = append3147186163529004513t_bool(zip_nat_bool(Xs,Us),zip_nat_bool(Ys,Vs)) ) ) ).

% zip_append
tff(fact_8184_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    linord4252657396651189596t_real(bot_bot_set_real) = nil_real ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_8185_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    linord193515670363874663t_bool(bot_bot_set_bool) = nil_bool ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_8186_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    linord2612477271533052124et_int(bot_bot_set_int) = nil_int ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_8187_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    aa_set_nat_list_nat(linord2614967742042102400et_nat,bot_bot_set_nat) = nil_nat ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_8188_horner__sum__simps_I1_J,axiom,
    ! [F: fun_nat_complex,A: complex] : groups404637655443745499omplex(F,A,nil_nat) = zero_zero_complex ).

% horner_sum_simps(1)
tff(fact_8189_horner__sum__simps_I1_J,axiom,
    ! [F: fun_int_complex,A: complex] : groups1380173120649922871omplex(F,A,nil_int) = zero_zero_complex ).

% horner_sum_simps(1)
tff(fact_8190_horner__sum__simps_I1_J,axiom,
    ! [F: fun_nat_real,A: real] : groups3482786445295563865t_real(F,A,nil_nat) = zero_zero_real ).

% horner_sum_simps(1)
tff(fact_8191_horner__sum__simps_I1_J,axiom,
    ! [F: fun_int_real,A: real] : groups5669708019988585653t_real(F,A,nil_int) = zero_zero_real ).

% horner_sum_simps(1)
tff(fact_8192_horner__sum__simps_I1_J,axiom,
    ! [F: fun_nat_rat,A: rat] : groups6853238114764508677at_rat(F,A,nil_nat) = zero_zero_rat ).

% horner_sum_simps(1)
tff(fact_8193_horner__sum__simps_I1_J,axiom,
    ! [F: fun_int_rat,A: rat] : groups7852591826665563233nt_rat(F,A,nil_int) = zero_zero_rat ).

% horner_sum_simps(1)
tff(fact_8194_horner__sum__simps_I1_J,axiom,
    ! [F: fun_nat_nat,A: nat] : aa_list_nat_nat(groups7488368174851004413at_nat(F,A),nil_nat) = zero_zero_nat ).

% horner_sum_simps(1)
tff(fact_8195_horner__sum__simps_I1_J,axiom,
    ! [F: fun_int_nat,A: nat] : groups8487721886752058969nt_nat(F,A,nil_int) = zero_zero_nat ).

% horner_sum_simps(1)
tff(fact_8196_horner__sum__simps_I1_J,axiom,
    ! [F: fun_nat_int,A: int] : groups7485877704341954137at_int(F,A,nil_nat) = zero_zero_int ).

% horner_sum_simps(1)
tff(fact_8197_horner__sum__simps_I1_J,axiom,
    ! [F: fun_int_int,A: int] : groups8485231416243008693nt_int(F,A,nil_int) = zero_zero_int ).

% horner_sum_simps(1)
tff(fact_8198_fun__upds__append__drop,axiom,
    ! [Xs: list_real,Ys: list_real,M: fun_real_option_real,Zs: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( map_upds_real_real(M,append_real(Xs,Zs),Ys) = map_upds_real_real(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8199_fun__upds__append__drop,axiom,
    ! [Xs: list_real,Ys: list_bool,M: fun_real_option_bool,Zs: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( map_upds_real_bool(M,append_real(Xs,Zs),Ys) = map_upds_real_bool(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8200_fun__upds__append__drop,axiom,
    ! [Xs: list_real,Ys: list_nat,M: fun_real_option_nat,Zs: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( map_upds_real_nat(M,append_real(Xs,Zs),Ys) = map_upds_real_nat(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8201_fun__upds__append__drop,axiom,
    ! [Xs: list_real,Ys: list_int,M: fun_real_option_int,Zs: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( map_upds_real_int(M,append_real(Xs,Zs),Ys) = map_upds_real_int(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8202_fun__upds__append__drop,axiom,
    ! [Xs: list_bool,Ys: list_real,M: fun_bool_option_real,Zs: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( map_upds_bool_real(M,append_bool(Xs,Zs),Ys) = map_upds_bool_real(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8203_fun__upds__append__drop,axiom,
    ! [Xs: list_bool,Ys: list_bool,M: fun_bool_option_bool,Zs: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( map_upds_bool_bool(M,append_bool(Xs,Zs),Ys) = map_upds_bool_bool(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8204_fun__upds__append__drop,axiom,
    ! [Xs: list_bool,Ys: list_nat,M: fun_bool_option_nat,Zs: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( map_upds_bool_nat(M,append_bool(Xs,Zs),Ys) = map_upds_bool_nat(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8205_fun__upds__append__drop,axiom,
    ! [Xs: list_bool,Ys: list_int,M: fun_bool_option_int,Zs: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( map_upds_bool_int(M,append_bool(Xs,Zs),Ys) = map_upds_bool_int(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8206_fun__upds__append__drop,axiom,
    ! [Xs: list_nat,Ys: list_real,M: fun_nat_option_real,Zs: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( map_upds_nat_real(M,append_nat(Xs,Zs),Ys) = map_upds_nat_real(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8207_fun__upds__append__drop,axiom,
    ! [Xs: list_nat,Ys: list_bool,M: fun_nat_option_bool,Zs: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( map_upds_nat_bool(M,append_nat(Xs,Zs),Ys) = map_upds_nat_bool(M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_8208_fun__upds__append2__drop,axiom,
    ! [Xs: list_real,Ys: list_real,M: fun_real_option_real,Zs: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( map_upds_real_real(M,Xs,append_real(Ys,Zs)) = map_upds_real_real(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8209_fun__upds__append2__drop,axiom,
    ! [Xs: list_real,Ys: list_bool,M: fun_real_option_bool,Zs: list_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( map_upds_real_bool(M,Xs,append_bool(Ys,Zs)) = map_upds_real_bool(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8210_fun__upds__append2__drop,axiom,
    ! [Xs: list_real,Ys: list_nat,M: fun_real_option_nat,Zs: list_nat] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( map_upds_real_nat(M,Xs,append_nat(Ys,Zs)) = map_upds_real_nat(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8211_fun__upds__append2__drop,axiom,
    ! [Xs: list_real,Ys: list_int,M: fun_real_option_int,Zs: list_int] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( map_upds_real_int(M,Xs,append_int(Ys,Zs)) = map_upds_real_int(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8212_fun__upds__append2__drop,axiom,
    ! [Xs: list_bool,Ys: list_real,M: fun_bool_option_real,Zs: list_real] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( map_upds_bool_real(M,Xs,append_real(Ys,Zs)) = map_upds_bool_real(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8213_fun__upds__append2__drop,axiom,
    ! [Xs: list_bool,Ys: list_bool,M: fun_bool_option_bool,Zs: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( map_upds_bool_bool(M,Xs,append_bool(Ys,Zs)) = map_upds_bool_bool(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8214_fun__upds__append2__drop,axiom,
    ! [Xs: list_bool,Ys: list_nat,M: fun_bool_option_nat,Zs: list_nat] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( map_upds_bool_nat(M,Xs,append_nat(Ys,Zs)) = map_upds_bool_nat(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8215_fun__upds__append2__drop,axiom,
    ! [Xs: list_bool,Ys: list_int,M: fun_bool_option_int,Zs: list_int] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( map_upds_bool_int(M,Xs,append_int(Ys,Zs)) = map_upds_bool_int(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8216_fun__upds__append2__drop,axiom,
    ! [Xs: list_nat,Ys: list_real,M: fun_nat_option_real,Zs: list_real] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( map_upds_nat_real(M,Xs,append_real(Ys,Zs)) = map_upds_nat_real(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8217_fun__upds__append2__drop,axiom,
    ! [Xs: list_nat,Ys: list_bool,M: fun_nat_option_bool,Zs: list_bool] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( map_upds_nat_bool(M,Xs,append_bool(Ys,Zs)) = map_upds_nat_bool(M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_8218_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_nat_fun_nat_assn,L: list_nat] : vEBT_L8301102511889123557at_nat(P,L,nil_nat) = aa_bool_assn(pure_assn,aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L),nil_nat)) ).

% list_assn_aux_simps(2)
tff(fact_8219_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_int_fun_nat_assn,L: list_int] : vEBT_L77084186935402305nt_nat(P,L,nil_nat) = aa_bool_assn(pure_assn,aa_list_int_bool(fequal_list_int(L),nil_int)) ).

% list_assn_aux_simps(2)
tff(fact_8220_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_nat_fun_int_assn,L: list_nat] : vEBT_L8298612041380073281at_int(P,L,nil_int) = aa_bool_assn(pure_assn,aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L),nil_nat)) ).

% list_assn_aux_simps(2)
tff(fact_8221_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_int_fun_int_assn,L: list_int] : vEBT_L74593716426352029nt_int(P,L,nil_int) = aa_bool_assn(pure_assn,aa_list_int_bool(fequal_list_int(L),nil_int)) ).

% list_assn_aux_simps(2)
tff(fact_8222_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_VE905575822219544865i_assn,L: list_VEBT_VEBT] : aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P,L),nil_VEBT_VEBTi) = aa_bool_assn(pure_assn,fequal591674592636813711T_VEBT(L,nil_VEBT_VEBT)) ).

% list_assn_aux_simps(2)
tff(fact_8223_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_nat_fun_nat_assn,L4: list_nat] : vEBT_L8301102511889123557at_nat(P,nil_nat,L4) = aa_bool_assn(pure_assn,aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L4),nil_nat)) ).

% list_assn_aux_simps(1)
tff(fact_8224_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_nat_fun_int_assn,L4: list_int] : vEBT_L8298612041380073281at_int(P,nil_nat,L4) = aa_bool_assn(pure_assn,aa_list_int_bool(fequal_list_int(L4),nil_int)) ).

% list_assn_aux_simps(1)
tff(fact_8225_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_int_fun_nat_assn,L4: list_nat] : vEBT_L77084186935402305nt_nat(P,nil_int,L4) = aa_bool_assn(pure_assn,aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L4),nil_nat)) ).

% list_assn_aux_simps(1)
tff(fact_8226_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_int_fun_int_assn,L4: list_int] : vEBT_L74593716426352029nt_int(P,nil_int,L4) = aa_bool_assn(pure_assn,aa_list_int_bool(fequal_list_int(L4),nil_int)) ).

% list_assn_aux_simps(1)
tff(fact_8227_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_VE905575822219544865i_assn,L4: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P,nil_VEBT_VEBT),L4) = aa_bool_assn(pure_assn,fequal1389750275112723248_VEBTi(L4,nil_VEBT_VEBTi)) ).

% list_assn_aux_simps(1)
tff(fact_8228_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),S3))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_adk(A),B),S3) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),S3))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_adk(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_8229_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_adl(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_adl(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_8230_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_adm(A),B),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_adm(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_8231_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_adn(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_adn(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_8232_sum_Odelta,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A),S3))
         => ( groups8024822376189712711omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ado(A),B),S3) = aa_Cod5621350732091595593omplex(B,A) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A),S3))
         => ( groups8024822376189712711omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ado(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_8233_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),S3))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_adp(A),B),S3) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),S3))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_adp(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_8234_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_adq(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_adq(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_8235_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_adr(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_adr(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_8236_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ads(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ads(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_8237_sum_Odelta,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A),S3))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_adt(A),B),S3) = aa_Code_integer_real(B,A) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A),S3))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_adt(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_8238_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),S3))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_adu(A),B),S3) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),S3))
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_adu(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_8239_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_adv(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_adv(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_8240_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_adw(A),B),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_adw(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_8241_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_adx(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_adx(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_8242_sum_Odelta_H,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: fun_Co1900716531650368264omplex] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A),S3))
         => ( groups8024822376189712711omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ady(A),B),S3) = aa_Cod5621350732091595593omplex(B,A) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A),S3))
         => ( groups8024822376189712711omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ady(A),B),S3) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_8243_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S3))
     => ( ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),S3))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_adz(A),B),S3) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A),S3))
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_adz(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_8244_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S3))
     => ( ( pp(aa_set_real_bool(member_real(A),S3))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_aea(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S3))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_aea(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_8245_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_aeb(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_aeb(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_8246_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_aec(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_aec(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_8247_sum_Odelta_H,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: fun_Co3960896333048504454r_real] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S3))
     => ( ( pp(aa_set1374477347934578310r_bool(member_Code_integer(A),S3))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_aed(A),B),S3) = aa_Code_integer_real(B,A) ) )
        & ( ~ pp(aa_set1374477347934578310r_bool(member_Code_integer(A),S3))
         => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_aed(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_8248_sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adi(F)),A2))) ).

% sum_abs
tff(fact_8249_sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_set_int_int(groups4538972089207619220nt_int(F),A2))),aa_set_int_int(groups4538972089207619220nt_int(aTP_Lamm_adj(F)),A2))) ).

% sum_abs
tff(fact_8250_length__greater__0__conv,axiom,
    ! [Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_real_nat(size_size_list_real,Xs)))
    <=> ( Xs != nil_real ) ) ).

% length_greater_0_conv
tff(fact_8251_length__greater__0__conv,axiom,
    ! [Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_bool_nat(size_size_list_bool,Xs)))
    <=> ( Xs != nil_bool ) ) ).

% length_greater_0_conv
tff(fact_8252_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs)))
    <=> ( Xs != nil_nat ) ) ).

% length_greater_0_conv
tff(fact_8253_length__greater__0__conv,axiom,
    ! [Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_int_nat(size_size_list_int,Xs)))
    <=> ( Xs != nil_int ) ) ).

% length_greater_0_conv
tff(fact_8254_nth__append__first,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,L4: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(append_VEBT_VEBTi(L,L4)),I) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I) ) ) ).

% nth_append_first
tff(fact_8255_nth__append__first,axiom,
    ! [I: nat,L: list_VEBT_VEBT,L4: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(append_VEBT_VEBT(L,L4)),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I) ) ) ).

% nth_append_first
tff(fact_8256_nth__append__first,axiom,
    ! [I: nat,L: list_real,L4: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_real_nat(size_size_list_real,L)))
     => ( aa_nat_real(nth_real(append_real(L,L4)),I) = aa_nat_real(nth_real(L),I) ) ) ).

% nth_append_first
tff(fact_8257_nth__append__first,axiom,
    ! [I: nat,L: list_bool,L4: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_bool_nat(size_size_list_bool,L)))
     => ( pp(aa_nat_bool(nth_bool(append_bool(L,L4)),I))
      <=> pp(aa_nat_bool(nth_bool(L),I)) ) ) ).

% nth_append_first
tff(fact_8258_nth__append__first,axiom,
    ! [I: nat,L: list_nat,L4: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)))
     => ( aa_nat_nat(nth_nat(append_nat(L,L4)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_append_first
tff(fact_8259_nth__append__first,axiom,
    ! [I: nat,L: list_int,L4: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_list_int_nat(size_size_list_int,L)))
     => ( aa_nat_int(nth_int(append_int(L,L4)),I) = aa_nat_int(nth_int(L),I) ) ) ).

% nth_append_first
tff(fact_8260_nth__append__length,axiom,
    ! [Xs: list_VEBT_VEBTi,X3: vEBT_VEBTi,Ys: list_VEBT_VEBTi] : aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(append_VEBT_VEBTi(Xs,cons_VEBT_VEBTi(X3,Ys))),size_s7982070591426661849_VEBTi(Xs)) = X3 ).

% nth_append_length
tff(fact_8261_nth__append__length,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT,Ys: list_VEBT_VEBT] : aa_nat_VEBT_VEBT(nth_VEBT_VEBT(append_VEBT_VEBT(Xs,cons_VEBT_VEBT(X3,Ys))),size_s6755466524823107622T_VEBT(Xs)) = X3 ).

% nth_append_length
tff(fact_8262_nth__append__length,axiom,
    ! [Xs: list_real,X3: real,Ys: list_real] : aa_nat_real(nth_real(append_real(Xs,cons_real(X3,Ys))),aa_list_real_nat(size_size_list_real,Xs)) = X3 ).

% nth_append_length
tff(fact_8263_nth__append__length,axiom,
    ! [Xs: list_bool,X3: bool,Ys: list_bool] :
      ( pp(aa_nat_bool(nth_bool(append_bool(Xs,cons_bool(X3,Ys))),aa_list_bool_nat(size_size_list_bool,Xs)))
    <=> pp(X3) ) ).

% nth_append_length
tff(fact_8264_nth__append__length,axiom,
    ! [Xs: list_nat,X3: nat,Ys: list_nat] : aa_nat_nat(nth_nat(append_nat(Xs,aa_list_nat_list_nat(cons_nat(X3),Ys))),aa_list_nat_nat(size_size_list_nat,Xs)) = X3 ).

% nth_append_length
tff(fact_8265_nth__append__length,axiom,
    ! [Xs: list_int,X3: int,Ys: list_int] : aa_nat_int(nth_int(append_int(Xs,aa_list_int_list_int(cons_int(X3),Ys))),aa_list_int_nat(size_size_list_int,Xs)) = X3 ).

% nth_append_length
tff(fact_8266_nth__append__length__plus,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi,N: nat] : aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(append_VEBT_VEBTi(Xs,Ys)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s7982070591426661849_VEBTi(Xs)),N)) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),N) ).

% nth_append_length_plus
tff(fact_8267_nth__append__length__plus,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,N: nat] : aa_nat_VEBT_VEBT(nth_VEBT_VEBT(append_VEBT_VEBT(Xs,Ys)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6755466524823107622T_VEBT(Xs)),N)) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),N) ).

% nth_append_length_plus
tff(fact_8268_nth__append__length__plus,axiom,
    ! [Xs: list_real,Ys: list_real,N: nat] : aa_nat_real(nth_real(append_real(Xs,Ys)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_real_nat(size_size_list_real,Xs)),N)) = aa_nat_real(nth_real(Ys),N) ).

% nth_append_length_plus
tff(fact_8269_nth__append__length__plus,axiom,
    ! [Xs: list_bool,Ys: list_bool,N: nat] :
      ( pp(aa_nat_bool(nth_bool(append_bool(Xs,Ys)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_bool_nat(size_size_list_bool,Xs)),N)))
    <=> pp(aa_nat_bool(nth_bool(Ys),N)) ) ).

% nth_append_length_plus
tff(fact_8270_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] : aa_nat_nat(nth_nat(append_nat(Xs,Ys)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(size_size_list_nat,Xs)),N)) = aa_nat_nat(nth_nat(Ys),N) ).

% nth_append_length_plus
tff(fact_8271_nth__append__length__plus,axiom,
    ! [Xs: list_int,Ys: list_int,N: nat] : aa_nat_int(nth_int(append_int(Xs,Ys)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_int_nat(size_size_list_int,Xs)),N)) = aa_nat_int(nth_int(Ys),N) ).

% nth_append_length_plus
tff(fact_8272_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( nil_real = drop_real(N,Xs) )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_8273_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( nil_bool = drop_bool(N,Xs) )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_8274_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( nil_nat = drop_nat(N,Xs) )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_8275_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( nil_int = drop_int(N,Xs) )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_8276_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( drop_real(N,Xs) = nil_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_8277_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_bool] :
      ( ( drop_bool(N,Xs) = nil_bool )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_8278_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( drop_nat(N,Xs) = nil_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_8279_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( drop_int(N,Xs) = nil_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_8280_drop__all,axiom,
    ! [Xs: list_real,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Xs)),N))
     => ( drop_real(N,Xs) = nil_real ) ) ).

% drop_all
tff(fact_8281_drop__all,axiom,
    ! [Xs: list_bool,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Xs)),N))
     => ( drop_bool(N,Xs) = nil_bool ) ) ).

% drop_all
tff(fact_8282_drop__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),N))
     => ( drop_nat(N,Xs) = nil_nat ) ) ).

% drop_all
tff(fact_8283_drop__all,axiom,
    ! [Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Xs)),N))
     => ( drop_int(N,Xs) = nil_int ) ) ).

% drop_all
tff(fact_8284_list__update__length,axiom,
    ! [Xs: list_VEBT_VEBTi,X3: vEBT_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] : list_u6098035379799741383_VEBTi(append_VEBT_VEBTi(Xs,cons_VEBT_VEBTi(X3,Ys)),size_s7982070591426661849_VEBTi(Xs),Y) = append_VEBT_VEBTi(Xs,cons_VEBT_VEBTi(Y,Ys)) ).

% list_update_length
tff(fact_8285_list__update__length,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] : aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(append_VEBT_VEBT(Xs,cons_VEBT_VEBT(X3,Ys)),size_s6755466524823107622T_VEBT(Xs)),Y) = append_VEBT_VEBT(Xs,cons_VEBT_VEBT(Y,Ys)) ).

% list_update_length
tff(fact_8286_list__update__length,axiom,
    ! [Xs: list_real,X3: real,Ys: list_real,Y: real] : aa_real_list_real(list_update_real(append_real(Xs,cons_real(X3,Ys)),aa_list_real_nat(size_size_list_real,Xs)),Y) = append_real(Xs,cons_real(Y,Ys)) ).

% list_update_length
tff(fact_8287_list__update__length,axiom,
    ! [Xs: list_bool,X3: bool,Ys: list_bool,Y: bool] : list_update_bool(append_bool(Xs,cons_bool(X3,Ys)),aa_list_bool_nat(size_size_list_bool,Xs),Y) = append_bool(Xs,cons_bool(Y,Ys)) ).

% list_update_length
tff(fact_8288_list__update__length,axiom,
    ! [Xs: list_nat,X3: nat,Ys: list_nat,Y: nat] : list_update_nat(append_nat(Xs,aa_list_nat_list_nat(cons_nat(X3),Ys)),aa_list_nat_nat(size_size_list_nat,Xs),Y) = append_nat(Xs,aa_list_nat_list_nat(cons_nat(Y),Ys)) ).

% list_update_length
tff(fact_8289_list__update__length,axiom,
    ! [Xs: list_int,X3: int,Ys: list_int,Y: int] : list_update_int(append_int(Xs,aa_list_int_list_int(cons_int(X3),Ys)),aa_list_int_nat(size_size_list_int,Xs),Y) = append_int(Xs,aa_list_int_list_int(cons_int(Y),Ys)) ).

% list_update_length
tff(fact_8290_take__append,axiom,
    ! [N: nat,Xs: list_real,Ys: list_real] : take_real(N,append_real(Xs,Ys)) = append_real(take_real(N,Xs),take_real(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_list_real_nat(size_size_list_real,Xs)),Ys)) ).

% take_append
tff(fact_8291_take__append,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_bool] : take_bool(N,append_bool(Xs,Ys)) = append_bool(take_bool(N,Xs),take_bool(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)),Ys)) ).

% take_append
tff(fact_8292_take__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] : take_nat(N,append_nat(Xs,Ys)) = append_nat(take_nat(N,Xs),take_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)),Ys)) ).

% take_append
tff(fact_8293_take__append,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int] : take_int(N,append_int(Xs,Ys)) = append_int(take_int(N,Xs),take_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_list_int_nat(size_size_list_int,Xs)),Ys)) ).

% take_append
tff(fact_8294_drop__append,axiom,
    ! [N: nat,Xs: list_real,Ys: list_real] : drop_real(N,append_real(Xs,Ys)) = append_real(drop_real(N,Xs),drop_real(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_list_real_nat(size_size_list_real,Xs)),Ys)) ).

% drop_append
tff(fact_8295_drop__append,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_bool] : drop_bool(N,append_bool(Xs,Ys)) = append_bool(drop_bool(N,Xs),drop_bool(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)),Ys)) ).

% drop_append
tff(fact_8296_drop__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] : drop_nat(N,append_nat(Xs,Ys)) = append_nat(drop_nat(N,Xs),drop_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)),Ys)) ).

% drop_append
tff(fact_8297_drop__append,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int] : drop_int(N,append_int(Xs,Ys)) = append_int(drop_int(N,Xs),drop_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_list_int_nat(size_size_list_int,Xs)),Ys)) ).

% drop_append
tff(fact_8298_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( ( linord2324613341767563021nteger(A2) = nil_Code_integer )
      <=> ( A2 = bot_bo3990330152332043303nteger ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_8299_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( linord4252657396651189596t_real(A2) = nil_real )
      <=> ( A2 = bot_bot_set_real ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_8300_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_bool] :
      ( pp(aa_set_bool_bool(finite_finite_bool,A2))
     => ( ( linord193515670363874663t_bool(A2) = nil_bool )
      <=> ( A2 = bot_bot_set_bool ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_8301_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( linord2612477271533052124et_int(A2) = nil_int )
      <=> ( A2 = bot_bot_set_int ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_8302_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_list_nat(linord2614967742042102400et_nat,A2) = nil_nat )
      <=> ( A2 = bot_bot_set_nat ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_8303_list__assn__aux__append2,axiom,
    ! [L22: list_real,L24: list_real,P: fun_re6303261851654292474l_assn,L1: list_real,L13: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,L22) = aa_list_real_nat(size_size_list_real,L24) )
     => ( vEBT_L1930518968523514909l_real(P,append_real(L1,L22),append_real(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1930518968523514909l_real(P,L1,L13)),vEBT_L1930518968523514909l_real(P,L22,L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8304_list__assn__aux__append2,axiom,
    ! [L22: list_real,L24: list_bool,P: fun_re2206215847901373695l_assn,L1: list_real,L13: list_bool] :
      ( ( aa_list_real_nat(size_size_list_real,L22) = aa_list_bool_nat(size_size_list_bool,L24) )
     => ( vEBT_L870199270341830246l_bool(P,append_real(L1,L22),append_bool(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L870199270341830246l_bool(P,L1,L13)),vEBT_L870199270341830246l_bool(P,L22,L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8305_list__assn__aux__append2,axiom,
    ! [L22: list_real,L24: list_nat,P: fun_re574865750389657758t_assn,L1: list_real,L13: list_nat] :
      ( ( aa_list_real_nat(size_size_list_real,L22) = aa_list_nat_nat(size_size_list_nat,L24) )
     => ( vEBT_L1446010312343316929al_nat(P,append_real(L1,L22),append_nat(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1446010312343316929al_nat(P,L1,L13)),vEBT_L1446010312343316929al_nat(P,L22,L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8306_list__assn__aux__append2,axiom,
    ! [L22: list_real,L24: list_int,P: fun_re4990660043736809722t_assn,L1: list_real,L13: list_int] :
      ( ( aa_list_real_nat(size_size_list_real,L22) = aa_list_int_nat(size_size_list_int,L24) )
     => ( vEBT_L1443519841834266653al_int(P,append_real(L1,L22),append_int(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1443519841834266653al_int(P,L1,L13)),vEBT_L1443519841834266653al_int(P,L22,L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8307_list__assn__aux__append2,axiom,
    ! [L22: list_bool,L24: list_real,P: fun_bo8885878543415060347l_assn,L1: list_bool,L13: list_real] :
      ( ( aa_list_bool_nat(size_size_list_bool,L22) = aa_list_real_nat(size_size_list_real,L24) )
     => ( vEBT_L5107238702033023424l_real(P,append_bool(L1,L22),append_real(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L5107238702033023424l_real(P,L1,L13)),vEBT_L5107238702033023424l_real(P,L22,L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8308_list__assn__aux__append2,axiom,
    ! [L22: list_bool,L24: list_bool,P: fun_bo8172996160782635326l_assn,L1: list_bool,L13: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,L22) = aa_list_bool_nat(size_size_list_bool,L24) )
     => ( aa_list_bool_assn(vEBT_L6562251693209944067l_bool(P,append_bool(L1,L22)),append_bool(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_list_bool_assn(vEBT_L6562251693209944067l_bool(P,L1),L13)),aa_list_bool_assn(vEBT_L6562251693209944067l_bool(P,L22),L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8309_list__assn__aux__append2,axiom,
    ! [L22: list_bool,L24: list_nat,P: fun_bo8438563643002758303t_assn,L1: list_bool,L13: list_nat] :
      ( ( aa_list_bool_nat(size_size_list_bool,L22) = aa_list_nat_nat(size_size_list_nat,L24) )
     => ( aa_list_nat_assn(vEBT_L7520539408223738084ol_nat(P,append_bool(L1,L22)),append_nat(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_list_nat_assn(vEBT_L7520539408223738084ol_nat(P,L1),L13)),aa_list_nat_assn(vEBT_L7520539408223738084ol_nat(P,L22),L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8310_list__assn__aux__append2,axiom,
    ! [L22: list_bool,L24: list_int,P: fun_bo3630985899495134459t_assn,L1: list_bool,L13: list_int] :
      ( ( aa_list_bool_nat(size_size_list_bool,L22) = aa_list_int_nat(size_size_list_int,L24) )
     => ( vEBT_L7518048937714687808ol_int(P,append_bool(L1,L22),append_int(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7518048937714687808ol_int(P,L1,L13)),vEBT_L7518048937714687808ol_int(P,L22,L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8311_list__assn__aux__append2,axiom,
    ! [L22: list_nat,L24: list_real,P: fun_na448437756743965086l_assn,L1: list_nat,L13: list_real] :
      ( ( aa_list_nat_nat(size_size_list_nat,L22) = aa_list_real_nat(size_size_list_real,L24) )
     => ( vEBT_L6102073776069194049t_real(P,append_nat(L1,L22),append_real(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6102073776069194049t_real(P,L1,L13)),vEBT_L6102073776069194049t_real(P,L22,L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8312_list__assn__aux__append2,axiom,
    ! [L22: list_nat,L24: list_bool,P: fun_na5136419689362302555l_assn,L1: list_nat,L13: list_bool] :
      ( ( aa_list_nat_nat(size_size_list_nat,L22) = aa_list_bool_nat(size_size_list_bool,L24) )
     => ( vEBT_L2559740421116301378t_bool(P,append_nat(L1,L22),append_bool(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2559740421116301378t_bool(P,L1,L13)),vEBT_L2559740421116301378t_bool(P,L22,L24)) ) ) ).

% list_assn_aux_append2
tff(fact_8313_list__assn__aux__append,axiom,
    ! [L1: list_real,L13: list_real,P: fun_re6303261851654292474l_assn,L22: list_real,L24: list_real] :
      ( ( aa_list_real_nat(size_size_list_real,L1) = aa_list_real_nat(size_size_list_real,L13) )
     => ( vEBT_L1930518968523514909l_real(P,append_real(L1,L22),append_real(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1930518968523514909l_real(P,L1,L13)),vEBT_L1930518968523514909l_real(P,L22,L24)) ) ) ).

% list_assn_aux_append
tff(fact_8314_list__assn__aux__append,axiom,
    ! [L1: list_real,L13: list_bool,P: fun_re2206215847901373695l_assn,L22: list_real,L24: list_bool] :
      ( ( aa_list_real_nat(size_size_list_real,L1) = aa_list_bool_nat(size_size_list_bool,L13) )
     => ( vEBT_L870199270341830246l_bool(P,append_real(L1,L22),append_bool(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L870199270341830246l_bool(P,L1,L13)),vEBT_L870199270341830246l_bool(P,L22,L24)) ) ) ).

% list_assn_aux_append
tff(fact_8315_list__assn__aux__append,axiom,
    ! [L1: list_real,L13: list_nat,P: fun_re574865750389657758t_assn,L22: list_real,L24: list_nat] :
      ( ( aa_list_real_nat(size_size_list_real,L1) = aa_list_nat_nat(size_size_list_nat,L13) )
     => ( vEBT_L1446010312343316929al_nat(P,append_real(L1,L22),append_nat(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1446010312343316929al_nat(P,L1,L13)),vEBT_L1446010312343316929al_nat(P,L22,L24)) ) ) ).

% list_assn_aux_append
tff(fact_8316_list__assn__aux__append,axiom,
    ! [L1: list_real,L13: list_int,P: fun_re4990660043736809722t_assn,L22: list_real,L24: list_int] :
      ( ( aa_list_real_nat(size_size_list_real,L1) = aa_list_int_nat(size_size_list_int,L13) )
     => ( vEBT_L1443519841834266653al_int(P,append_real(L1,L22),append_int(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1443519841834266653al_int(P,L1,L13)),vEBT_L1443519841834266653al_int(P,L22,L24)) ) ) ).

% list_assn_aux_append
tff(fact_8317_list__assn__aux__append,axiom,
    ! [L1: list_bool,L13: list_real,P: fun_bo8885878543415060347l_assn,L22: list_bool,L24: list_real] :
      ( ( aa_list_bool_nat(size_size_list_bool,L1) = aa_list_real_nat(size_size_list_real,L13) )
     => ( vEBT_L5107238702033023424l_real(P,append_bool(L1,L22),append_real(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L5107238702033023424l_real(P,L1,L13)),vEBT_L5107238702033023424l_real(P,L22,L24)) ) ) ).

% list_assn_aux_append
tff(fact_8318_list__assn__aux__append,axiom,
    ! [L1: list_bool,L13: list_bool,P: fun_bo8172996160782635326l_assn,L22: list_bool,L24: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,L1) = aa_list_bool_nat(size_size_list_bool,L13) )
     => ( aa_list_bool_assn(vEBT_L6562251693209944067l_bool(P,append_bool(L1,L22)),append_bool(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_list_bool_assn(vEBT_L6562251693209944067l_bool(P,L1),L13)),aa_list_bool_assn(vEBT_L6562251693209944067l_bool(P,L22),L24)) ) ) ).

% list_assn_aux_append
tff(fact_8319_list__assn__aux__append,axiom,
    ! [L1: list_bool,L13: list_nat,P: fun_bo8438563643002758303t_assn,L22: list_bool,L24: list_nat] :
      ( ( aa_list_bool_nat(size_size_list_bool,L1) = aa_list_nat_nat(size_size_list_nat,L13) )
     => ( aa_list_nat_assn(vEBT_L7520539408223738084ol_nat(P,append_bool(L1,L22)),append_nat(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_list_nat_assn(vEBT_L7520539408223738084ol_nat(P,L1),L13)),aa_list_nat_assn(vEBT_L7520539408223738084ol_nat(P,L22),L24)) ) ) ).

% list_assn_aux_append
tff(fact_8320_list__assn__aux__append,axiom,
    ! [L1: list_bool,L13: list_int,P: fun_bo3630985899495134459t_assn,L22: list_bool,L24: list_int] :
      ( ( aa_list_bool_nat(size_size_list_bool,L1) = aa_list_int_nat(size_size_list_int,L13) )
     => ( vEBT_L7518048937714687808ol_int(P,append_bool(L1,L22),append_int(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7518048937714687808ol_int(P,L1,L13)),vEBT_L7518048937714687808ol_int(P,L22,L24)) ) ) ).

% list_assn_aux_append
tff(fact_8321_list__assn__aux__append,axiom,
    ! [L1: list_nat,L13: list_real,P: fun_na448437756743965086l_assn,L22: list_nat,L24: list_real] :
      ( ( aa_list_nat_nat(size_size_list_nat,L1) = aa_list_real_nat(size_size_list_real,L13) )
     => ( vEBT_L6102073776069194049t_real(P,append_nat(L1,L22),append_real(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6102073776069194049t_real(P,L1,L13)),vEBT_L6102073776069194049t_real(P,L22,L24)) ) ) ).

% list_assn_aux_append
tff(fact_8322_list__assn__aux__append,axiom,
    ! [L1: list_nat,L13: list_bool,P: fun_na5136419689362302555l_assn,L22: list_nat,L24: list_bool] :
      ( ( aa_list_nat_nat(size_size_list_nat,L1) = aa_list_bool_nat(size_size_list_bool,L13) )
     => ( vEBT_L2559740421116301378t_bool(P,append_nat(L1,L22),append_bool(L13,L24)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2559740421116301378t_bool(P,L1,L13)),vEBT_L2559740421116301378t_bool(P,L22,L24)) ) ) ).

% list_assn_aux_append
tff(fact_8323_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adi(F)),A2))) ).

% sum_abs_ge_zero
tff(fact_8324_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_set_int_int(groups4538972089207619220nt_int(aTP_Lamm_adj(F)),A2))) ).

% sum_abs_ge_zero
tff(fact_8325_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( n_lists_nat(N,nil_nat) = cons_list_nat(nil_nat,nil_list_nat) ) )
      & ( ( N != zero_zero_nat )
       => ( n_lists_nat(N,nil_nat) = nil_list_nat ) ) ) ).

% n_lists_Nil
tff(fact_8326_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( n_lists_int(N,nil_int) = cons_list_int(nil_int,nil_list_int) ) )
      & ( ( N != zero_zero_nat )
       => ( n_lists_int(N,nil_int) = nil_list_int ) ) ) ).

% n_lists_Nil
tff(fact_8327_length__ge__1__conv,axiom,
    ! [L: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_list_real_nat(size_size_list_real,L)))
    <=> ( L != nil_real ) ) ).

% length_ge_1_conv
tff(fact_8328_length__ge__1__conv,axiom,
    ! [L: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_list_bool_nat(size_size_list_bool,L)))
    <=> ( L != nil_bool ) ) ).

% length_ge_1_conv
tff(fact_8329_length__ge__1__conv,axiom,
    ! [L: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_list_nat_nat(size_size_list_nat,L)))
    <=> ( L != nil_nat ) ) ).

% length_ge_1_conv
tff(fact_8330_length__ge__1__conv,axiom,
    ! [L: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_list_int_nat(size_size_list_int,L)))
    <=> ( L != nil_int ) ) ).

% length_ge_1_conv
tff(fact_8331_concat__map__singleton,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] : concat_real(map_VE755361874264402988t_real(aTP_Lamm_aee(F),Xs)) = map_VEBT_VEBT_real(F,Xs) ).

% concat_map_singleton
tff(fact_8332_concat__map__singleton,axiom,
    ! [F: fun_Pr4298194452408323251l_bool,Xs: list_P2722043638750504429l_bool] : concat_bool(map_Pr8255831621680831418t_bool(aTP_Lamm_aef(F),Xs)) = map_Pr8574941368599290932l_bool(F,Xs) ).

% concat_map_singleton
tff(fact_8333_concat__map__singleton,axiom,
    ! [F: fun_nat_bool,Xs: list_nat] : concat_bool(map_nat_list_bool(aTP_Lamm_aeg(F),Xs)) = map_nat_bool(F,Xs) ).

% concat_map_singleton
tff(fact_8334_concat__map__singleton,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] : concat_nat(map_VE3614494202555519440st_nat(aTP_Lamm_aeh(F),Xs)) = map_VEBT_VEBT_nat(F,Xs) ).

% concat_map_singleton
tff(fact_8335_concat__map__singleton,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] : concat_nat(map_nat_list_nat(aTP_Lamm_aei(F),Xs)) = map_nat_nat(F,Xs) ).

% concat_map_singleton
tff(fact_8336_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups2073611262835488442omplex(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N))),aa_nat_complex(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8337_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N))),aa_nat_rat(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8338_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N))),aa_nat_int(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8339_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N))),aa_nat_nat(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8340_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N))),aa_nat_real(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8341_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_aej(C),A2) = aa_nat_rat(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_aej(C),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_8342_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_aek(C),A2) = aa_nat_complex(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_aek(C),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_8343_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ael(C)),A2) = aa_nat_real(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ael(C)),A2) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_8344_map__upds__append1,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBTi,M: fun_re869386379898482478_VEBTi,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,Xs)),size_s7982070591426661849_VEBTi(Ys)))
     => ( map_up4277394084060657369_VEBTi(M,append_real(Xs,cons_real(X3,nil_real)),Ys) = fun_up5845057532826650672_VEBTi(map_up4277394084060657369_VEBTi(M,Xs,Ys),X3,some_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),aa_list_real_nat(size_size_list_real,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8345_map__upds__append1,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,M: fun_re6149904099307231015T_VEBT,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( map_up3774473461586936848T_VEBT(M,append_real(Xs,cons_real(X3,nil_real)),Ys) = fun_up3388931106328796303T_VEBT(map_up3774473461586936848T_VEBT(M,Xs,Ys),X3,some_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),aa_list_real_nat(size_size_list_real,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8346_map__upds__append1,axiom,
    ! [Xs: list_real,Ys: list_num,M: fun_real_option_num,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,Xs)),size_size_list_num(Ys)))
     => ( map_upds_real_num(M,append_real(Xs,cons_real(X3,nil_real)),Ys) = fun_up7385324149885497365on_num(map_upds_real_num(M,Xs,Ys),X3,aa_num_option_num(some_num,aa_nat_num(nth_num(Ys),aa_list_real_nat(size_size_list_real,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8347_map__upds__append1,axiom,
    ! [Xs: list_real,Ys: list_real,M: fun_real_option_real,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_real_nat(size_size_list_real,Ys)))
     => ( map_upds_real_real(M,append_real(Xs,cons_real(X3,nil_real)),Ys) = fun_up7779911115344551847n_real(map_upds_real_real(M,Xs,Ys),X3,aa_real_option_real(some_real,aa_nat_real(nth_real(Ys),aa_list_real_nat(size_size_list_real,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8348_map__upds__append1,axiom,
    ! [Xs: list_real,Ys: list_bool,M: fun_real_option_bool,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_bool_nat(size_size_list_bool,Ys)))
     => ( map_upds_real_bool(M,append_real(Xs,cons_real(X3,nil_real)),Ys) = fun_up251216825372557554n_bool(map_upds_real_bool(M,Xs,Ys),X3,aa_bool_option_bool(some_bool,aa_nat_bool(nth_bool(Ys),aa_list_real_nat(size_size_list_real,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8349_map__upds__append1,axiom,
    ! [Xs: list_real,Ys: list_nat,M: fun_real_option_nat,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_nat_nat(size_size_list_nat,Ys)))
     => ( map_upds_real_nat(M,append_real(Xs,cons_real(X3,nil_real)),Ys) = fun_up6677080212936659659on_nat(map_upds_real_nat(M,Xs,Ys),X3,aa_nat_option_nat(some_nat,aa_nat_nat(nth_nat(Ys),aa_list_real_nat(size_size_list_real,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8350_map__upds__append1,axiom,
    ! [Xs: list_real,Ys: list_int,M: fun_real_option_int,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_real_nat(size_size_list_real,Xs)),aa_list_int_nat(size_size_list_int,Ys)))
     => ( map_upds_real_int(M,append_real(Xs,cons_real(X3,nil_real)),Ys) = fun_up2499229193427462951on_int(map_upds_real_int(M,Xs,Ys),X3,aa_int_option_int(some_int,aa_nat_int(nth_int(Ys),aa_list_real_nat(size_size_list_real,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8351_map__upds__append1,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBTi,M: fun_bo5052482999933817007_VEBTi,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_bool_nat(size_size_list_bool,Xs)),size_s7982070591426661849_VEBTi(Ys)))
     => ( map_up3574140571279353302_VEBTi(M,append_bool(Xs,cons_bool(X3,nil_bool)),Ys) = fun_up7125570336889275103_VEBTi(map_up3574140571279353302_VEBTi(M,Xs,Ys),X3,some_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),aa_list_bool_nat(size_size_list_bool,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8352_map__upds__append1,axiom,
    ! [Xs: list_bool,Ys: list_VEBT_VEBT,M: fun_bo8890943138197189862T_VEBT,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_bool_nat(size_size_list_bool,Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( map_up5237471190051075923T_VEBT(M,append_bool(Xs,cons_bool(X3,nil_bool)),Ys) = fun_up5192449144741904032T_VEBT(map_up5237471190051075923T_VEBT(M,Xs,Ys),X3,some_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),aa_list_bool_nat(size_size_list_bool,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8353_map__upds__append1,axiom,
    ! [Xs: list_bool,Ys: list_num,M: fun_bool_option_num,X3: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_list_bool_nat(size_size_list_bool,Xs)),size_size_list_num(Ys)))
     => ( map_upds_bool_num(M,append_bool(Xs,cons_bool(X3,nil_bool)),Ys) = fun_up5214249621878093252on_num(map_upds_bool_num(M,Xs,Ys),X3,aa_num_option_num(some_num,aa_nat_num(nth_num(Ys),aa_list_bool_nat(size_size_list_bool,Xs)))) ) ) ).

% map_upds_append1
tff(fact_8354_same__length__different,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( Xs != Ys )
     => ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
       => ? [Pre: list_real,X: real,Xs5: list_real,Y3: real,Ys4: list_real] :
            ( ( X != Y3 )
            & ( Xs = append_real(Pre,append_real(cons_real(X,nil_real),Xs5)) )
            & ( Ys = append_real(Pre,append_real(cons_real(Y3,nil_real),Ys4)) ) ) ) ) ).

% same_length_different
tff(fact_8355_same__length__different,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( Xs != Ys )
     => ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
       => ? [Pre: list_bool,X: bool,Xs5: list_bool,Y3: bool,Ys4: list_bool] :
            ( ~ ( pp(X)
              <=> pp(Y3) )
            & ( Xs = append_bool(Pre,append_bool(cons_bool(X,nil_bool),Xs5)) )
            & ( Ys = append_bool(Pre,append_bool(cons_bool(Y3,nil_bool),Ys4)) ) ) ) ) ).

% same_length_different
tff(fact_8356_same__length__different,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) )
       => ? [Pre: list_nat,X: nat,Xs5: list_nat,Y3: nat,Ys4: list_nat] :
            ( ( X != Y3 )
            & ( Xs = append_nat(Pre,append_nat(aa_list_nat_list_nat(cons_nat(X),nil_nat),Xs5)) )
            & ( Ys = append_nat(Pre,append_nat(aa_list_nat_list_nat(cons_nat(Y3),nil_nat),Ys4)) ) ) ) ) ).

% same_length_different
tff(fact_8357_same__length__different,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs != Ys )
     => ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
       => ? [Pre: list_int,X: int,Xs5: list_int,Y3: int,Ys4: list_int] :
            ( ( X != Y3 )
            & ( Xs = append_int(Pre,append_int(aa_list_int_list_int(cons_int(X),nil_int),Xs5)) )
            & ( Ys = append_int(Pre,append_int(aa_list_int_list_int(cons_int(Y3),nil_int),Ys4)) ) ) ) ) ).

% same_length_different
tff(fact_8358_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B3: set_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_aem(G),B3)),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_aeo(G),A2)),B3) ).

% sum.swap
tff(fact_8359_sum_Oswap,axiom,
    ! [G: fun_co7673397065115134084omplex,B3: set_complex,A2: set_complex] : groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_aep(G),B3),A2) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_aer(G),A2),B3) ).

% sum.swap
tff(fact_8360_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_real,B3: set_nat,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_aes(G),B3)),A2) = aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_aeu(G),A2)),B3) ).

% sum.swap
tff(fact_8361_sum_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B3: set_int,A2: set_int] : aa_set_int_int(groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_aev(G),B3)),A2) = aa_set_int_int(groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_aex(G),A2)),B3) ).

% sum.swap
tff(fact_8362_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( groups2073611262835488442omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8363_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VE5569070535936029429omplex,A2: set_VEBT_VEBT] :
      ( ( groups1794756597179926696omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A2))
           => ( aa_VEBT_VEBT_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8364_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups5754745047067104278omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8365_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups3049146728041665814omplex(G,A2) != zero_zero_complex )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_complex(G,A3) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8366_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_real,A2: set_VEBT_VEBT] :
      ( ( groups2240296850493347238T_real(G,A2) != zero_zero_real )
     => ~ ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A2))
           => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8367_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8368_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8369_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8370_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_rat,A2: set_VEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(A3),A2))
           => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8371_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups1300246762558778688al_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: real] :
            ( pp(aa_set_real_bool(member_real(A3),A2))
           => ( aa_real_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8372_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => ( aa_nat_nat(G,X) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_8373_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X: complex] :
          ( pp(aa_set_complex_bool(member_complex(X),A2))
         => ( aa_complex_complex(G,X) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_8374_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => ( aa_nat_real(G,X) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_8375_sum_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => ( aa_int_int(G,X) = zero_zero_int ) )
     => ( aa_set_int_int(groups4538972089207619220nt_int(G),A2) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_8376_map__by__foldl,axiom,
    ! [F: fun_VEBT_VEBT_real,L: list_VEBT_VEBT] : foldl_4796148479939130645T_VEBT(aTP_Lamm_aey(F),nil_real,L) = map_VEBT_VEBT_real(F,L) ).

% map_by_foldl
tff(fact_8377_map__by__foldl,axiom,
    ! [F: fun_Pr4298194452408323251l_bool,L: list_P2722043638750504429l_bool] : foldl_8414172701497714859l_bool(aTP_Lamm_aez(F),nil_bool,L) = map_Pr8574941368599290932l_bool(F,L) ).

% map_by_foldl
tff(fact_8378_map__by__foldl,axiom,
    ! [F: fun_nat_bool,L: list_nat] : foldl_list_bool_nat(aTP_Lamm_afa(F),nil_bool,L) = map_nat_bool(F,L) ).

% map_by_foldl
tff(fact_8379_map__by__foldl,axiom,
    ! [F: fun_VEBT_VEBT_nat,L: list_VEBT_VEBT] : foldl_7309499613077172081T_VEBT(aTP_Lamm_afb(F),nil_nat,L) = map_VEBT_VEBT_nat(F,L) ).

% map_by_foldl
tff(fact_8380_map__by__foldl,axiom,
    ! [F: fun_nat_nat,L: list_nat] : foldl_list_nat_nat(aTP_Lamm_afc(F),nil_nat,L) = map_nat_nat(F,L) ).

% map_by_foldl
tff(fact_8381_subset__eq__mset__impl_Ocases,axiom,
    ! [X3: produc1828647624359046049st_nat] :
      ( ! [Ys5: list_nat] : X3 != produc2694037385005941721st_nat(nil_nat,Ys5)
     => ~ ! [X: nat,Xs2: list_nat,Ys5: list_nat] : X3 != produc2694037385005941721st_nat(aa_list_nat_list_nat(cons_nat(X),Xs2),Ys5) ) ).

% subset_eq_mset_impl.cases
tff(fact_8382_subset__eq__mset__impl_Ocases,axiom,
    ! [X3: produc1186641810826059865st_int] :
      ( ! [Ys5: list_int] : X3 != produc364263696895485585st_int(nil_int,Ys5)
     => ~ ! [X: int,Xs2: list_int,Ys5: list_int] : X3 != produc364263696895485585st_int(aa_list_int_list_int(cons_int(X),Xs2),Ys5) ) ).

% subset_eq_mset_impl.cases
tff(fact_8383_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5))) ) ).

% sum_mono
tff(fact_8384_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I2)),aa_VEBT_VEBT_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,K5)),groups136491112297645522BT_rat(G,K5))) ) ).

% sum_mono
tff(fact_8385_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5))) ) ).

% sum_mono
tff(fact_8386_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5))) ) ).

% sum_mono
tff(fact_8387_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I2)),aa_VEBT_VEBT_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,K5)),groups771621172384141258BT_nat(G,K5))) ) ).

% sum_mono
tff(fact_8388_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5))) ) ).

% sum_mono
tff(fact_8389_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5))) ) ).

% sum_mono
tff(fact_8390_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5))) ) ).

% sum_mono
tff(fact_8391_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( ! [I2: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_VEBT_VEBT_int(F,I2)),aa_VEBT_VEBT_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups769130701875090982BT_int(F,K5)),groups769130701875090982BT_int(G,K5))) ) ).

% sum_mono
tff(fact_8392_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( pp(aa_set_real_bool(member_real(I2),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5))) ) ).

% sum_mono
tff(fact_8393_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_mm(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_8394_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_nc(G),H),A2) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_8395_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_me(G),H)),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),A2)),aa_set_nat_real(groups6591440286371151544t_real(H),A2)) ).

% sum.distrib
tff(fact_8396_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : aa_set_int_int(groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_afd(G),H)),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_int_int(groups4538972089207619220nt_int(G),A2)),aa_set_int_int(groups4538972089207619220nt_int(H),A2)) ).

% sum.distrib
tff(fact_8397_sum__product,axiom,
    ! [F: fun_nat_nat,A2: set_nat,G: fun_nat_nat,B3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_aff(F),G),B3)),A2) ).

% sum_product
tff(fact_8398_sum__product,axiom,
    ! [F: fun_complex_complex,A2: set_complex,G: fun_complex_complex,B3: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,B3)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_afh(F),G),B3),A2) ).

% sum_product
tff(fact_8399_sum__product,axiom,
    ! [F: fun_nat_real,A2: set_nat,G: fun_nat_real,B3: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)),aa_set_nat_real(groups6591440286371151544t_real(G),B3)) = aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_afj(F),G),B3)),A2) ).

% sum_product
tff(fact_8400_sum__product,axiom,
    ! [F: fun_int_int,A2: set_int,G: fun_int_int,B3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_int_int(groups4538972089207619220nt_int(F),A2)),aa_set_int_int(groups4538972089207619220nt_int(G),B3)) = aa_set_int_int(groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_afl(F),G),B3)),A2) ).

% sum_product
tff(fact_8401_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A2: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_afm(F),R2)),A2) ).

% sum_distrib_right
tff(fact_8402_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A2)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_afn(F),R2),A2) ).

% sum_distrib_right
tff(fact_8403_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A2: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)),R2) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_afo(F),R2)),A2) ).

% sum_distrib_right
tff(fact_8404_sum__distrib__right,axiom,
    ! [F: fun_int_int,A2: set_int,R2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_int_int(groups4538972089207619220nt_int(F),A2)),R2) = aa_set_int_int(groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_afp(F),R2)),A2) ).

% sum_distrib_right
tff(fact_8405_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_afq(R2),F)),A2) ).

% sum_distrib_left
tff(fact_8406_sum__distrib__left,axiom,
    ! [R2: complex,F: fun_complex_complex,A2: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),groups7754918857620584856omplex(F,A2)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_afr(R2),F),A2) ).

% sum_distrib_left
tff(fact_8407_sum__distrib__left,axiom,
    ! [R2: real,F: fun_nat_real,A2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_set_nat_real(groups6591440286371151544t_real(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_afs(R2),F)),A2) ).

% sum_distrib_left
tff(fact_8408_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A2: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_set_int_int(groups4538972089207619220nt_int(F),A2)) = aa_set_int_int(groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aft(R2),F)),A2) ).

% sum_distrib_left
tff(fact_8409_sum__subtractf,axiom,
    ! [F: fun_complex_complex,G: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_afu(F),G),A2) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,A2)) ).

% sum_subtractf
tff(fact_8410_sum__subtractf,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_afv(F),G)),A2) = aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(F),A2)),aa_set_nat_real(groups6591440286371151544t_real(G),A2)) ).

% sum_subtractf
tff(fact_8411_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A2: set_int] : aa_set_int_int(groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_afw(F),G)),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_int_int(groups4538972089207619220nt_int(F),A2)),aa_set_int_int(groups4538972089207619220nt_int(G),A2)) ).

% sum_subtractf
tff(fact_8412_sum__negf,axiom,
    ! [F: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_afx(F),A2) = aa_complex_complex(uminus1482373934393186551omplex,groups7754918857620584856omplex(F,A2)) ).

% sum_negf
tff(fact_8413_sum__negf,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_afy(F)),A2) = aa_real_real(uminus_uminus_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)) ).

% sum_negf
tff(fact_8414_sum__negf,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_set_int_int(groups4538972089207619220nt_int(aTP_Lamm_afz(F)),A2) = aa_int_int(uminus_uminus_int,aa_set_int_int(groups4538972089207619220nt_int(F),A2)) ).

% sum_negf
tff(fact_8415_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_nat,G: fun_VE597728521808591204at_nat,R: fun_VE3763956218935105199t_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( groups771621172384141258BT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_agb(B3),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_agd(A2),G),R)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8416_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_nat,G: fun_real_fun_nat_nat,R: fun_re9121486324101951821t_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( groups1935376822645274424al_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_agf(B3),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_agh(A2),G),R)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8417_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B3: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( groups4541462559716669496nt_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_agj(B3),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_agm(A2),G),R)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8418_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B3: set_nat,G: fun_co636942507509095880at_nat,R: fun_co2841405903858895691t_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( groups5693394587270226106ex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_ago(B3),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_agr(A2),G),R)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8419_sum_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B3: set_nat,G: fun_Co3312552843764780407at_nat,R: fun_Co8657364630979331932t_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( groups7237345082560585321er_nat(aa_fun2949368542467568330er_nat(aa_fun8366171557965857828er_nat(aTP_Lamm_agt(B3),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7549014261834767933at_nat(aa_fun5721305349739569687at_nat(aTP_Lamm_agw(A2),G),R)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8420_sum_Oswap__restrict,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_complex,G: fun_VE2006399705683203104omplex,R: fun_VE2128610149277289041x_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
       => ( groups1794756597179926696omplex(aa_fun6203732500129703078omplex(aa_fun7613540521737166425omplex(aTP_Lamm_agy(B3),G),R),A2) = groups7754918857620584856omplex(aa_fun1095524714264483688omplex(aa_fun8876071380179997707omplex(aTP_Lamm_aha(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8421_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_complex,G: fun_re7208826350674946562omplex,R: fun_re3010880487650781423x_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
       => ( groups5754745047067104278omplex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_ahc(B3),G),R),A2) = groups7754918857620584856omplex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_ahe(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8422_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B3: set_complex,G: fun_na1591161823130732966omplex,R: fun_na5941084329111710283x_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
       => ( groups2073611262835488442omplex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_ahg(B3),G),R),A2) = groups7754918857620584856omplex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_ahj(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8423_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B3: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5750378413879446255x_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
       => ( groups3049146728041665814omplex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_ahl(B3),G),R),A2) = groups7754918857620584856omplex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_aho(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8424_sum_Oswap__restrict,axiom,
    ! [A2: set_Code_integer,B3: set_complex,G: fun_Co8905303033605938227omplex,R: fun_Co7124838579184476414x_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B3))
       => ( groups8024822376189712711omplex(aa_fun7983975303565649482omplex(aa_fun843835629537646824omplex(aTP_Lamm_ahq(B3),G),R),A2) = groups7754918857620584856omplex(aa_fun8169706760919072411omplex(aa_fun2334956106851614393omplex(aTP_Lamm_aht(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_8425_length__Suc__rev__conv,axiom,
    ! [Xs: list_real,N: nat] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Ys3: list_real,Y5: real] :
          ( ( Xs = append_real(Ys3,cons_real(Y5,nil_real)) )
          & ( aa_list_real_nat(size_size_list_real,Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_8426_length__Suc__rev__conv,axiom,
    ! [Xs: list_bool,N: nat] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Ys3: list_bool,Y5: bool] :
          ( ( Xs = append_bool(Ys3,cons_bool(Y5,nil_bool)) )
          & ( aa_list_bool_nat(size_size_list_bool,Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_8427_length__Suc__rev__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Ys3: list_nat,Y5: nat] :
          ( ( Xs = append_nat(Ys3,aa_list_nat_list_nat(cons_nat(Y5),nil_nat)) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_8428_length__Suc__rev__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Ys3: list_int,Y5: int] :
          ( ( Xs = append_int(Ys3,aa_list_int_list_int(cons_int(Y5),nil_int)) )
          & ( aa_list_int_nat(size_size_list_int,Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_8429_length__Suc__conv__rev,axiom,
    ! [Xs: list_real,N: nat] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: real,Ys3: list_real] :
          ( ( Xs = append_real(Ys3,cons_real(Y5,nil_real)) )
          & ( aa_list_real_nat(size_size_list_real,Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_8430_length__Suc__conv__rev,axiom,
    ! [Xs: list_bool,N: nat] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: bool,Ys3: list_bool] :
          ( ( Xs = append_bool(Ys3,cons_bool(Y5,nil_bool)) )
          & ( aa_list_bool_nat(size_size_list_bool,Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_8431_length__Suc__conv__rev,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: nat,Ys3: list_nat] :
          ( ( Xs = append_nat(Ys3,aa_list_nat_list_nat(cons_nat(Y5),nil_nat)) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_8432_length__Suc__conv__rev,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y5: int,Ys3: list_int] :
          ( ( Xs = append_int(Ys3,aa_list_int_list_int(cons_int(Y5),nil_int)) )
          & ( aa_list_int_nat(size_size_list_int,Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_8433_length__append__singleton,axiom,
    ! [Xs: list_real,X3: real] : aa_list_real_nat(size_size_list_real,append_real(Xs,cons_real(X3,nil_real))) = aa_nat_nat(suc,aa_list_real_nat(size_size_list_real,Xs)) ).

% length_append_singleton
tff(fact_8434_length__append__singleton,axiom,
    ! [Xs: list_bool,X3: bool] : aa_list_bool_nat(size_size_list_bool,append_bool(Xs,cons_bool(X3,nil_bool))) = aa_nat_nat(suc,aa_list_bool_nat(size_size_list_bool,Xs)) ).

% length_append_singleton
tff(fact_8435_length__append__singleton,axiom,
    ! [Xs: list_nat,X3: nat] : aa_list_nat_nat(size_size_list_nat,append_nat(Xs,aa_list_nat_list_nat(cons_nat(X3),nil_nat))) = aa_nat_nat(suc,aa_list_nat_nat(size_size_list_nat,Xs)) ).

% length_append_singleton
tff(fact_8436_length__append__singleton,axiom,
    ! [Xs: list_int,X3: int] : aa_list_int_nat(size_size_list_int,append_int(Xs,aa_list_int_list_int(cons_int(X3),nil_int))) = aa_nat_nat(suc,aa_list_int_nat(size_size_list_int,Xs)) ).

% length_append_singleton
tff(fact_8437_length__compl__rev__induct,axiom,
    ! [P: fun_list_real_bool,L: list_real] :
      ( pp(aa_list_real_bool(P,nil_real))
     => ( ! [L2: list_real,E: real] :
            ( ! [Ll: list_real] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Ll)),aa_list_real_nat(size_size_list_real,L2)))
               => pp(aa_list_real_bool(P,Ll)) )
           => pp(aa_list_real_bool(P,append_real(L2,cons_real(E,nil_real)))) )
       => pp(aa_list_real_bool(P,L)) ) ) ).

% length_compl_rev_induct
tff(fact_8438_length__compl__rev__induct,axiom,
    ! [P: fun_list_bool_bool,L: list_bool] :
      ( pp(aa_list_bool_bool(P,nil_bool))
     => ( ! [L2: list_bool,E: bool] :
            ( ! [Ll: list_bool] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Ll)),aa_list_bool_nat(size_size_list_bool,L2)))
               => pp(aa_list_bool_bool(P,Ll)) )
           => pp(aa_list_bool_bool(P,append_bool(L2,cons_bool(E,nil_bool)))) )
       => pp(aa_list_bool_bool(P,L)) ) ) ).

% length_compl_rev_induct
tff(fact_8439_length__compl__rev__induct,axiom,
    ! [P: fun_list_nat_bool,L: list_nat] :
      ( pp(aa_list_nat_bool(P,nil_nat))
     => ( ! [L2: list_nat,E: nat] :
            ( ! [Ll: list_nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Ll)),aa_list_nat_nat(size_size_list_nat,L2)))
               => pp(aa_list_nat_bool(P,Ll)) )
           => pp(aa_list_nat_bool(P,append_nat(L2,aa_list_nat_list_nat(cons_nat(E),nil_nat)))) )
       => pp(aa_list_nat_bool(P,L)) ) ) ).

% length_compl_rev_induct
tff(fact_8440_length__compl__rev__induct,axiom,
    ! [P: fun_list_int_bool,L: list_int] :
      ( pp(aa_list_int_bool(P,nil_int))
     => ( ! [L2: list_int,E: int] :
            ( ! [Ll: list_int] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Ll)),aa_list_int_nat(size_size_list_int,L2)))
               => pp(aa_list_int_bool(P,Ll)) )
           => pp(aa_list_int_bool(P,append_int(L2,aa_list_int_list_int(cons_int(E),nil_int)))) )
       => pp(aa_list_int_bool(P,L)) ) ) ).

% length_compl_rev_induct
tff(fact_8441_replicate__Suc__conv__snoc,axiom,
    ! [N: nat,X3: nat] : replicate_nat(aa_nat_nat(suc,N),X3) = append_nat(replicate_nat(N,X3),aa_list_nat_list_nat(cons_nat(X3),nil_nat)) ).

% replicate_Suc_conv_snoc
tff(fact_8442_replicate__Suc__conv__snoc,axiom,
    ! [N: nat,X3: int] : replicate_int(aa_nat_nat(suc,N),X3) = append_int(replicate_int(N,X3),aa_list_int_list_int(cons_int(X3),nil_int)) ).

% replicate_Suc_conv_snoc
tff(fact_8443_replicate__Suc__conv__snoc,axiom,
    ! [N: nat,X3: vEBT_VEBT] : aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(suc,N)),X3) = append_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(N),X3),cons_VEBT_VEBT(X3,nil_VEBT_VEBT)) ).

% replicate_Suc_conv_snoc
tff(fact_8444_replicate__Suc__conv__snoc,axiom,
    ! [N: nat,X3: bool] : replicate_bool(aa_nat_nat(suc,N),X3) = append_bool(replicate_bool(N,X3),cons_bool(X3,nil_bool)) ).

% replicate_Suc_conv_snoc
tff(fact_8445_comm__append__is__replicate,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( Xs != nil_bool )
     => ( ( Ys != nil_bool )
       => ( ( append_bool(Xs,Ys) = append_bool(Ys,Xs) )
         => ? [N2: nat,Zs2: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N2))
              & ( concat_bool(replicate_list_bool(N2,Zs2)) = append_bool(Xs,Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_8446_comm__append__is__replicate,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_nat )
       => ( ( append_nat(Xs,Ys) = append_nat(Ys,Xs) )
         => ? [N2: nat,Zs2: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N2))
              & ( concat_nat(replicate_list_nat(N2,Zs2)) = append_nat(Xs,Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_8447_comm__append__is__replicate,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs != nil_int )
     => ( ( Ys != nil_int )
       => ( ( append_int(Xs,Ys) = append_int(Ys,Xs) )
         => ? [N2: nat,Zs2: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N2))
              & ( concat_int(replicate_list_int(N2,Zs2)) = append_int(Xs,Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_8448_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_real,Ys: list_real] : enumerate_real(N,append_real(Xs,Ys)) = append6678681742291297912t_real(enumerate_real(N,Xs),enumerate_real(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_list_real_nat(size_size_list_real,Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_8449_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_bool,Ys: list_bool] : enumerate_bool(N,append_bool(Xs,Ys)) = append3147186163529004513t_bool(enumerate_bool(N,Xs),enumerate_bool(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_list_bool_nat(size_size_list_bool,Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_8450_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] : enumerate_nat(N,append_nat(Xs,Ys)) = append985823374593552924at_nat(enumerate_nat(N,Xs),enumerate_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_list_nat_nat(size_size_list_nat,Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_8451_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int] : enumerate_int(N,append_int(Xs,Ys)) = append6031344391939132024at_int(enumerate_int(N,Xs),enumerate_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_list_int_nat(size_size_list_int,Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_8452_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] : n_lists_nat(zero_zero_nat,Xs) = cons_list_nat(nil_nat,nil_list_nat) ).

% n_lists.simps(1)
tff(fact_8453_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_int] : n_lists_int(zero_zero_nat,Xs) = cons_list_int(nil_int,nil_list_int) ).

% n_lists.simps(1)
tff(fact_8454_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2240296850493347238T_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_8455_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_8456_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_8457_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8458_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8459_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8460_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8461_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups771621172384141258BT_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_8462_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_8463_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_8464_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups2240296850493347238T_real(F,A2))) ) ).

% sum_nonneg
tff(fact_8465_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2))) ) ).

% sum_nonneg
tff(fact_8466_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2))) ) ).

% sum_nonneg
tff(fact_8467_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( pp(aa_set_nat_bool(member_nat(X),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8468_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups136491112297645522BT_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8469_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8470_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8471_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups771621172384141258BT_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_8472_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X: real] :
          ( pp(aa_set_real_bool(member_real(X),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_8473_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X: int] :
          ( pp(aa_set_int_bool(member_int(X),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_8474_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] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I2)),aa_VEBT_VEBT_rat(G,I2))) )
       => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I5))
           => ( aa_VEBT_VEBT_rat(F,I) = aa_VEBT_VEBT_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8475_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] :
            ( pp(aa_set_real_bool(member_real(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) )
       => ( pp(aa_set_real_bool(member_real(I),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8476_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] :
            ( pp(aa_set_nat_bool(member_nat(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) )
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8477_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] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8478_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] :
            ( pp(aa_set_complex_bool(member_complex(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8479_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] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Code_integer_rat(F,I2)),aa_Code_integer_rat(G,I2))) )
       => ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),I5))
         => ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,I5))
           => ( aa_Code_integer_rat(F,I) = aa_Code_integer_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8480_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] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I2)),aa_VEBT_VEBT_nat(G,I2))) )
       => ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(I),I5))
         => ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,I5))
           => ( aa_VEBT_VEBT_nat(F,I) = aa_VEBT_VEBT_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8481_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] :
            ( pp(aa_set_real_bool(member_real(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) )
       => ( pp(aa_set_real_bool(member_real(I),I5))
         => ( pp(aa_set_real_bool(finite_finite_real,I5))
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8482_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] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8483_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] :
            ( pp(aa_set_complex_bool(member_complex(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8484_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X)),A2))
           => ( 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),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8485_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X)),A2))
           => ( aa_nat_real(F,aa_nat_nat(suc,X)) = aa_nat_real(G,aa_nat_nat(suc,X)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A2) = aa_set_nat_real(groups6591440286371151544t_real(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8486_sorted__append__bigger,axiom,
    ! [Xs: list_real,Y: real] :
      ( pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),Xs))
     => ( ! [X: real] :
            ( pp(aa_set_real_bool(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) )
       => pp(aa_list_real_bool(sorted_wrt_real(ord_less_eq_real),append_real(Xs,cons_real(Y,nil_real)))) ) ) ).

% sorted_append_bigger
tff(fact_8487_sorted__append__bigger,axiom,
    ! [Xs: list_bool,Y: bool] :
      ( pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),Xs))
     => ( ! [X: bool] :
            ( pp(aa_set_bool_bool(member_bool(X),aa_lis3587183946281582166t_bool(set_bool2,Xs)))
           => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,X),Y)) )
       => pp(aa_list_bool_bool(sorted_wrt_bool(ord_less_eq_bool),append_bool(Xs,cons_bool(Y,nil_bool)))) ) ) ).

% sorted_append_bigger
tff(fact_8488_sorted__append__bigger,axiom,
    ! [Xs: list_rat,Y: rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => ( ! [X: rat] :
            ( pp(aa_set_rat_bool(member_rat(X),set_rat2(Xs)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) )
       => sorted_wrt_rat(ord_less_eq_rat,append_rat(Xs,cons_rat(Y,nil_rat))) ) ) ).

% sorted_append_bigger
tff(fact_8489_sorted__append__bigger,axiom,
    ! [Xs: list_num,Y: num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => ( ! [X: num] :
            ( pp(aa_set_num_bool(member_num(X),set_num2(Xs)))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) )
       => sorted_wrt_num(ord_less_eq_num,append_num(Xs,cons_num(Y,nil_num))) ) ) ).

% sorted_append_bigger
tff(fact_8490_sorted__append__bigger,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => ( ! [X: nat] :
            ( pp(aa_set_nat_bool(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) )
       => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),append_nat(Xs,aa_list_nat_list_nat(cons_nat(Y),nil_nat)))) ) ) ).

% sorted_append_bigger
tff(fact_8491_sorted__append__bigger,axiom,
    ! [Xs: list_int,Y: int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => ( ! [X: int] :
            ( pp(aa_set_int_bool(member_int(X),aa_list_int_set_int(set_int2,Xs)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) )
       => pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),append_int(Xs,aa_list_int_list_int(cons_int(Y),nil_int)))) ) ) ).

% sorted_append_bigger
tff(fact_8492_zip__append1,axiom,
    ! [Xs: list_real,Ys: list_real,Zs: list_nat] : zip_real_nat(append_real(Xs,Ys),Zs) = append2703634050814417656al_nat(zip_real_nat(Xs,take_nat(aa_list_real_nat(size_size_list_real,Xs),Zs)),zip_real_nat(Ys,drop_nat(aa_list_real_nat(size_size_list_real,Xs),Zs))) ).

% zip_append1
tff(fact_8493_zip__append1,axiom,
    ! [Xs: list_real,Ys: list_real,Zs: list_bool] : zip_real_bool(append_real(Xs,Ys),Zs) = append7873561321238512645l_bool(zip_real_bool(Xs,take_bool(aa_list_real_nat(size_size_list_real,Xs),Zs)),zip_real_bool(Ys,drop_bool(aa_list_real_nat(size_size_list_real,Xs),Zs))) ).

% zip_append1
tff(fact_8494_zip__append1,axiom,
    ! [Xs: list_bool,Ys: list_bool,Zs: list_nat] : zip_bool_nat(append_bool(Xs,Ys),Zs) = append4937290130733399867ol_nat(zip_bool_nat(Xs,take_nat(aa_list_bool_nat(size_size_list_bool,Xs),Zs)),zip_bool_nat(Ys,drop_nat(aa_list_bool_nat(size_size_list_bool,Xs),Zs))) ).

% zip_append1
tff(fact_8495_zip__append1,axiom,
    ! [Xs: list_bool,Ys: list_bool,Zs: list_bool] : zip_bool_bool(append_bool(Xs,Ys),Zs) = append1767198414729310338l_bool(zip_bool_bool(Xs,take_bool(aa_list_bool_nat(size_size_list_bool,Xs),Zs)),zip_bool_bool(Ys,drop_bool(aa_list_bool_nat(size_size_list_bool,Xs),Zs))) ).

% zip_append1
tff(fact_8496_zip__append1,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] : zip_nat_nat(append_nat(Xs,Ys),Zs) = append985823374593552924at_nat(zip_nat_nat(Xs,take_nat(aa_list_nat_nat(size_size_list_nat,Xs),Zs)),zip_nat_nat(Ys,drop_nat(aa_list_nat_nat(size_size_list_nat,Xs),Zs))) ).

% zip_append1
tff(fact_8497_zip__append1,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_bool] : zip_nat_bool(append_nat(Xs,Ys),Zs) = append3147186163529004513t_bool(zip_nat_bool(Xs,take_bool(aa_list_nat_nat(size_size_list_nat,Xs),Zs)),zip_nat_bool(Ys,drop_bool(aa_list_nat_nat(size_size_list_nat,Xs),Zs))) ).

% zip_append1
tff(fact_8498_zip__append1,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_nat] : zip_int_nat(append_int(Xs,Ys),Zs) = append1985177086494607480nt_nat(zip_int_nat(Xs,take_nat(aa_list_int_nat(size_size_list_int,Xs),Zs)),zip_int_nat(Ys,drop_nat(aa_list_int_nat(size_size_list_int,Xs),Zs))) ).

% zip_append1
tff(fact_8499_zip__append1,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_bool] : zip_int_bool(append_int(Xs,Ys),Zs) = append22895197060380293t_bool(zip_int_bool(Xs,take_bool(aa_list_int_nat(size_size_list_int,Xs),Zs)),zip_int_bool(Ys,drop_bool(aa_list_int_nat(size_size_list_int,Xs),Zs))) ).

% zip_append1
tff(fact_8500_zip__append2,axiom,
    ! [Xs: list_nat,Ys: list_real,Zs: list_real] : zip_nat_real(Xs,append_real(Ys,Zs)) = append6678681742291297912t_real(zip_nat_real(take_nat(aa_list_real_nat(size_size_list_real,Ys),Xs),Ys),zip_nat_real(drop_nat(aa_list_real_nat(size_size_list_real,Ys),Xs),Zs)) ).

% zip_append2
tff(fact_8501_zip__append2,axiom,
    ! [Xs: list_bool,Ys: list_real,Zs: list_real] : zip_bool_real(Xs,append_real(Ys,Zs)) = append8258889960777667607l_real(zip_bool_real(take_bool(aa_list_real_nat(size_size_list_real,Ys),Xs),Ys),zip_bool_real(drop_bool(aa_list_real_nat(size_size_list_real,Ys),Xs),Zs)) ).

% zip_append2
tff(fact_8502_zip__append2,axiom,
    ! [Xs: list_nat,Ys: list_bool,Zs: list_bool] : zip_nat_bool(Xs,append_bool(Ys,Zs)) = append3147186163529004513t_bool(zip_nat_bool(take_nat(aa_list_bool_nat(size_size_list_bool,Ys),Xs),Ys),zip_nat_bool(drop_nat(aa_list_bool_nat(size_size_list_bool,Ys),Xs),Zs)) ).

% zip_append2
tff(fact_8503_zip__append2,axiom,
    ! [Xs: list_bool,Ys: list_bool,Zs: list_bool] : zip_bool_bool(Xs,append_bool(Ys,Zs)) = append1767198414729310338l_bool(zip_bool_bool(take_bool(aa_list_bool_nat(size_size_list_bool,Ys),Xs),Ys),zip_bool_bool(drop_bool(aa_list_bool_nat(size_size_list_bool,Ys),Xs),Zs)) ).

% zip_append2
tff(fact_8504_zip__append2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] : zip_nat_nat(Xs,append_nat(Ys,Zs)) = append985823374593552924at_nat(zip_nat_nat(take_nat(aa_list_nat_nat(size_size_list_nat,Ys),Xs),Ys),zip_nat_nat(drop_nat(aa_list_nat_nat(size_size_list_nat,Ys),Xs),Zs)) ).

% zip_append2
tff(fact_8505_zip__append2,axiom,
    ! [Xs: list_bool,Ys: list_nat,Zs: list_nat] : zip_bool_nat(Xs,append_nat(Ys,Zs)) = append4937290130733399867ol_nat(zip_bool_nat(take_bool(aa_list_nat_nat(size_size_list_nat,Ys),Xs),Ys),zip_bool_nat(drop_bool(aa_list_nat_nat(size_size_list_nat,Ys),Xs),Zs)) ).

% zip_append2
tff(fact_8506_zip__append2,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_int] : zip_nat_int(Xs,append_int(Ys,Zs)) = append6031344391939132024at_int(zip_nat_int(take_nat(aa_list_int_nat(size_size_list_int,Ys),Xs),Ys),zip_nat_int(drop_nat(aa_list_int_nat(size_size_list_int,Ys),Xs),Zs)) ).

% zip_append2
tff(fact_8507_zip__append2,axiom,
    ! [Xs: list_bool,Ys: list_int,Zs: list_int] : zip_bool_int(Xs,append_int(Ys,Zs)) = append759439111224203159ol_int(zip_bool_int(take_bool(aa_list_int_nat(size_size_list_int,Ys),Xs),Ys),zip_bool_int(drop_bool(aa_list_int_nat(size_size_list_int,Ys),Xs),Zs)) ).

% zip_append2
tff(fact_8508_list__rest__coinc,axiom,
    ! [S22: list_real,S1: list_real,R12: list_real,R23: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,S22)),aa_list_real_nat(size_size_list_real,S1)))
     => ( ( append_real(S1,R12) = append_real(S22,R23) )
       => ? [R1p: list_real] : R23 = append_real(R1p,R12) ) ) ).

% list_rest_coinc
tff(fact_8509_list__rest__coinc,axiom,
    ! [S22: list_bool,S1: list_bool,R12: list_bool,R23: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,S22)),aa_list_bool_nat(size_size_list_bool,S1)))
     => ( ( append_bool(S1,R12) = append_bool(S22,R23) )
       => ? [R1p: list_bool] : R23 = append_bool(R1p,R12) ) ) ).

% list_rest_coinc
tff(fact_8510_list__rest__coinc,axiom,
    ! [S22: list_nat,S1: list_nat,R12: list_nat,R23: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,S22)),aa_list_nat_nat(size_size_list_nat,S1)))
     => ( ( append_nat(S1,R12) = append_nat(S22,R23) )
       => ? [R1p: list_nat] : R23 = append_nat(R1p,R12) ) ) ).

% list_rest_coinc
tff(fact_8511_list__rest__coinc,axiom,
    ! [S22: list_int,S1: list_int,R12: list_int,R23: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,S22)),aa_list_int_nat(size_size_list_int,S1)))
     => ( ( append_int(S1,R12) = append_int(S22,R23) )
       => ? [R1p: list_int] : R23 = append_int(R1p,R12) ) ) ).

% list_rest_coinc
tff(fact_8512_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( groups1794756597179926696omplex(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bg(A2),P))) = groups1794756597179926696omplex(aa_fun6792494735270171281omplex(aTP_Lamm_ahu(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8513_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bh(A2),P))) = groups5754745047067104278omplex(aa_fun6813637976755515871omplex(aTP_Lamm_ahv(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8514_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2073611262835488442omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_bl(A2),P))) = groups2073611262835488442omplex(aa_fun8588439219086927967omplex(aTP_Lamm_ahw(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8515_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bm(A2),P))) = groups3049146728041665814omplex(aa_fun3606824691168040799omplex(aTP_Lamm_ahx(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8516_sum_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: fun_Co1900716531650368264omplex,P: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( groups8024822376189712711omplex(G,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_ahy(A2),P))) = groups8024822376189712711omplex(aa_fun5444226037728322975omplex(aTP_Lamm_ahz(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8517_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
     => ( groups2240296850493347238T_real(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_bg(A2),P))) = groups2240296850493347238T_real(aa_fun2072160908118572431T_real(aTP_Lamm_aia(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8518_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_bh(A2),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_aib(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8519_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_bm(A2),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_aic(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8520_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_bi(A2),P))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_aid(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8521_sum_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real,P: fun_Co8822979462182099027r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( groups1270011288395367621r_real(G,collect_Code_integer(aa_fun8875953811088571772r_bool(aTP_Lamm_ahy(A2),P))) = groups1270011288395367621r_real(aa_fun1813952354503624093r_real(aTP_Lamm_aie(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8522_length__nth__simps_I1_J,axiom,
    aa_list_real_nat(size_size_list_real,nil_real) = zero_zero_nat ).

% length_nth_simps(1)
tff(fact_8523_length__nth__simps_I1_J,axiom,
    aa_list_bool_nat(size_size_list_bool,nil_bool) = zero_zero_nat ).

% length_nth_simps(1)
tff(fact_8524_length__nth__simps_I1_J,axiom,
    aa_list_nat_nat(size_size_list_nat,nil_nat) = zero_zero_nat ).

% length_nth_simps(1)
tff(fact_8525_length__nth__simps_I1_J,axiom,
    aa_list_int_nat(size_size_list_int,nil_int) = zero_zero_nat ).

% length_nth_simps(1)
tff(fact_8526_len__greater__imp__nonempty,axiom,
    ! [X3: nat,L: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_list_real_nat(size_size_list_real,L)))
     => ( L != nil_real ) ) ).

% len_greater_imp_nonempty
tff(fact_8527_len__greater__imp__nonempty,axiom,
    ! [X3: nat,L: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_list_bool_nat(size_size_list_bool,L)))
     => ( L != nil_bool ) ) ).

% len_greater_imp_nonempty
tff(fact_8528_len__greater__imp__nonempty,axiom,
    ! [X3: nat,L: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_list_nat_nat(size_size_list_nat,L)))
     => ( L != nil_nat ) ) ).

% len_greater_imp_nonempty
tff(fact_8529_len__greater__imp__nonempty,axiom,
    ! [X3: nat,L: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_list_int_nat(size_size_list_int,L)))
     => ( L != nil_int ) ) ).

% len_greater_imp_nonempty
tff(fact_8530_list__induct2,axiom,
    ! [Xs: list_real,Ys: list_real,P: fun_li4271802897377934289l_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(P,nil_real),nil_real))
       => ( ! [X: real,Xs2: list_real,Y3: real,Ys5: list_real] :
              ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
             => ( pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(P,Xs2),Ys5))
               => pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(P,cons_real(X,Xs2)),cons_real(Y3,Ys5))) ) )
         => pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8531_list__induct2,axiom,
    ! [Xs: list_real,Ys: list_bool,P: fun_li2895203809746299958l_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(P,nil_real),nil_bool))
       => ( ! [X: real,Xs2: list_real,Y3: bool,Ys5: list_bool] :
              ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_bool_nat(size_size_list_bool,Ys5) )
             => ( pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(P,Xs2),Ys5))
               => pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(P,cons_real(X,Xs2)),cons_bool(Y3,Ys5))) ) )
         => pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8532_list__induct2,axiom,
    ! [Xs: list_real,Ys: list_nat,P: fun_li804954861265393581t_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(P,nil_real),nil_nat))
       => ( ! [X: real,Xs2: list_real,Y3: nat,Ys5: list_nat] :
              ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_nat_nat(size_size_list_nat,Ys5) )
             => ( pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(P,Xs2),Ys5))
               => pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(P,cons_real(X,Xs2)),aa_list_nat_list_nat(cons_nat(Y3),Ys5))) ) )
         => pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8533_list__induct2,axiom,
    ! [Xs: list_real,Ys: list_int,P: fun_li3109360770246014033t_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( pp(aa_list_int_bool(aa_lis126974366075332282t_bool(P,nil_real),nil_int))
       => ( ! [X: real,Xs2: list_real,Y3: int,Ys5: list_int] :
              ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_int_nat(size_size_list_int,Ys5) )
             => ( pp(aa_list_int_bool(aa_lis126974366075332282t_bool(P,Xs2),Ys5))
               => pp(aa_list_int_bool(aa_lis126974366075332282t_bool(P,cons_real(X,Xs2)),aa_list_int_list_int(cons_int(Y3),Ys5))) ) )
         => pp(aa_list_int_bool(aa_lis126974366075332282t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8534_list__induct2,axiom,
    ! [Xs: list_bool,Ys: list_real,P: fun_li2322583063365907994l_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(P,nil_bool),nil_real))
       => ( ! [X: bool,Xs2: list_bool,Y3: real,Ys5: list_real] :
              ( ( aa_list_bool_nat(size_size_list_bool,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
             => ( pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(P,Xs2),Ys5))
               => pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(P,cons_bool(X,Xs2)),cons_real(Y3,Ys5))) ) )
         => pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8535_list__induct2,axiom,
    ! [Xs: list_bool,Ys: list_bool,P: fun_li402716648095076269l_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(P,nil_bool),nil_bool))
       => ( ! [X: bool,Xs2: list_bool,Y3: bool,Ys5: list_bool] :
              ( ( aa_list_bool_nat(size_size_list_bool,Xs2) = aa_list_bool_nat(size_size_list_bool,Ys5) )
             => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(P,Xs2),Ys5))
               => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(P,cons_bool(X,Xs2)),cons_bool(Y3,Ys5))) ) )
         => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8536_list__induct2,axiom,
    ! [Xs: list_bool,Ys: list_nat,P: fun_li1802912181200660086t_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(P,nil_bool),nil_nat))
       => ( ! [X: bool,Xs2: list_bool,Y3: nat,Ys5: list_nat] :
              ( ( aa_list_bool_nat(size_size_list_bool,Xs2) = aa_list_nat_nat(size_size_list_nat,Ys5) )
             => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(P,Xs2),Ys5))
               => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(P,cons_bool(X,Xs2)),aa_list_nat_list_nat(cons_nat(Y3),Ys5))) ) )
         => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8537_list__induct2,axiom,
    ! [Xs: list_bool,Ys: list_int,P: fun_li4107318090181280538t_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_list_int_nat(size_size_list_int,Ys) )
     => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(P,nil_bool),nil_int))
       => ( ! [X: bool,Xs2: list_bool,Y3: int,Ys5: list_int] :
              ( ( aa_list_bool_nat(size_size_list_bool,Xs2) = aa_list_int_nat(size_size_list_int,Ys5) )
             => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(P,Xs2),Ys5))
               => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(P,cons_bool(X,Xs2)),aa_list_int_list_int(cons_int(Y3),Ys5))) ) )
         => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8538_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_real,P: fun_li5924588734490278061l_bool] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(P,nil_nat),nil_real))
       => ( ! [X: nat,Xs2: list_nat,Y3: real,Ys5: list_real] :
              ( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
             => ( pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(P,Xs2),Ys5))
               => pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(P,aa_list_nat_list_nat(cons_nat(X),Xs2)),cons_real(Y3,Ys5))) ) )
         => pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8539_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_bool,P: fun_li6875962081873459290l_bool] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(P,nil_nat),nil_bool))
       => ( ! [X: nat,Xs2: list_nat,Y3: bool,Ys5: list_bool] :
              ( ( aa_list_nat_nat(size_size_list_nat,Xs2) = aa_list_bool_nat(size_size_list_bool,Ys5) )
             => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(P,Xs2),Ys5))
               => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(P,aa_list_nat_list_nat(cons_nat(X),Xs2)),cons_bool(Y3,Ys5))) ) )
         => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_8540_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_real,Zs: list_real,P: fun_li8357235125080761070l_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_real_nat(size_size_list_real,Zs) )
       => ( pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(aa_lis6354848994543922647l_bool(P,nil_real),nil_real),nil_real))
         => ( ! [X: real,Xs2: list_real,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
               => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_real_nat(size_size_list_real,Zs2) )
                 => ( pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(aa_lis6354848994543922647l_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(aa_lis6354848994543922647l_bool(P,cons_real(X,Xs2)),cons_real(Y3,Ys5)),cons_real(Z3,Zs2))) ) ) )
           => pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(aa_lis6354848994543922647l_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8541_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_real,Zs: list_bool,P: fun_li7985517392215786607l_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_bool_nat(size_size_list_bool,Zs) )
       => ( pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(aa_lis1658356894974515376l_bool(P,nil_real),nil_real),nil_bool))
         => ( ! [X: real,Xs2: list_real,Y3: real,Ys5: list_real,Z3: bool,Zs2: list_bool] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
               => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_bool_nat(size_size_list_bool,Zs2) )
                 => ( pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(aa_lis1658356894974515376l_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(aa_lis1658356894974515376l_bool(P,cons_real(X,Xs2)),cons_real(Y3,Ys5)),cons_bool(Z3,Zs2))) ) ) )
           => pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(aa_lis1658356894974515376l_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8542_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_real,Zs: list_nat,P: fun_li612048240254691786t_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_nat_nat(size_size_list_nat,Zs) )
       => ( pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(aa_lis2133206188042513587t_bool(P,nil_real),nil_real),nil_nat))
         => ( ! [X: real,Xs2: list_real,Y3: real,Ys5: list_real,Z3: nat,Zs2: list_nat] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
               => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_nat_nat(size_size_list_nat,Zs2) )
                 => ( pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(aa_lis2133206188042513587t_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(aa_lis2133206188042513587t_bool(P,cons_real(X,Xs2)),cons_real(Y3,Ys5)),aa_list_nat_list_nat(cons_nat(Z3),Zs2))) ) ) )
           => pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(aa_lis2133206188042513587t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8543_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_real,Zs: list_int,P: fun_li1298624273295242350t_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_int_nat(size_size_list_int,Zs) )
       => ( pp(aa_list_int_bool(aa_lis126974366075332282t_bool(aa_lis4437612097023134039t_bool(P,nil_real),nil_real),nil_int))
         => ( ! [X: real,Xs2: list_real,Y3: real,Ys5: list_real,Z3: int,Zs2: list_int] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
               => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_int_nat(size_size_list_int,Zs2) )
                 => ( pp(aa_list_int_bool(aa_lis126974366075332282t_bool(aa_lis4437612097023134039t_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_int_bool(aa_lis126974366075332282t_bool(aa_lis4437612097023134039t_bool(P,cons_real(X,Xs2)),cons_real(Y3,Ys5)),aa_list_int_list_int(cons_int(Z3),Zs2))) ) ) )
           => pp(aa_list_int_bool(aa_lis126974366075332282t_bool(aa_lis4437612097023134039t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8544_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_bool,Zs: list_real,P: fun_li4172524130268437587l_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( ( aa_list_bool_nat(size_size_list_bool,Ys) = aa_list_real_nat(size_size_list_real,Zs) )
       => ( pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(aa_lis1085736148594123412l_bool(P,nil_real),nil_bool),nil_real))
         => ( ! [X: real,Xs2: list_real,Y3: bool,Ys5: list_bool,Z3: real,Zs2: list_real] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_bool_nat(size_size_list_bool,Ys5) )
               => ( ( aa_list_bool_nat(size_size_list_bool,Ys5) = aa_list_real_nat(size_size_list_real,Zs2) )
                 => ( pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(aa_lis1085736148594123412l_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(aa_lis1085736148594123412l_bool(P,cons_real(X,Xs2)),cons_bool(Y3,Ys5)),cons_real(Z3,Zs2))) ) ) )
           => pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(aa_lis1085736148594123412l_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8545_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_bool,Zs: list_bool,P: fun_li4160648503519994314l_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( ( aa_list_bool_nat(size_size_list_bool,Ys) = aa_list_bool_nat(size_size_list_bool,Zs) )
       => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis1730967974872196275l_bool(P,nil_real),nil_bool),nil_bool))
         => ( ! [X: real,Xs2: list_real,Y3: bool,Ys5: list_bool,Z3: bool,Zs2: list_bool] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_bool_nat(size_size_list_bool,Ys5) )
               => ( ( aa_list_bool_nat(size_size_list_bool,Ys5) = aa_list_bool_nat(size_size_list_bool,Zs2) )
                 => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis1730967974872196275l_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis1730967974872196275l_bool(P,cons_real(X,Xs2)),cons_bool(Y3,Ys5)),cons_bool(Z3,Zs2))) ) ) )
           => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis1730967974872196275l_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8546_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_bool,Zs: list_nat,P: fun_li4129016536432936367t_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( ( aa_list_bool_nat(size_size_list_bool,Ys) = aa_list_nat_nat(size_size_list_nat,Zs) )
       => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis9131688112241455088t_bool(P,nil_real),nil_bool),nil_nat))
         => ( ! [X: real,Xs2: list_real,Y3: bool,Ys5: list_bool,Z3: nat,Zs2: list_nat] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_bool_nat(size_size_list_bool,Ys5) )
               => ( ( aa_list_bool_nat(size_size_list_bool,Ys5) = aa_list_nat_nat(size_size_list_nat,Zs2) )
                 => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis9131688112241455088t_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis9131688112241455088t_bool(P,cons_real(X,Xs2)),cons_bool(Y3,Ys5)),aa_list_nat_list_nat(cons_nat(Z3),Zs2))) ) ) )
           => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis9131688112241455088t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8547_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_bool,Zs: list_int,P: fun_li4815592569473486931t_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_bool_nat(size_size_list_bool,Ys) )
     => ( ( aa_list_bool_nat(size_size_list_bool,Ys) = aa_list_int_nat(size_size_list_int,Zs) )
       => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis2212721984367299732t_bool(P,nil_real),nil_bool),nil_int))
         => ( ! [X: real,Xs2: list_real,Y3: bool,Ys5: list_bool,Z3: int,Zs2: list_int] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_bool_nat(size_size_list_bool,Ys5) )
               => ( ( aa_list_bool_nat(size_size_list_bool,Ys5) = aa_list_int_nat(size_size_list_int,Zs2) )
                 => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis2212721984367299732t_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis2212721984367299732t_bool(P,cons_real(X,Xs2)),cons_bool(Y3,Ys5)),aa_list_int_list_int(cons_int(Z3),Zs2))) ) ) )
           => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis2212721984367299732t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8548_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_nat,Zs: list_real,P: fun_li5590782502790269642l_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( ( aa_list_nat_nat(size_size_list_nat,Ys) = aa_list_real_nat(size_size_list_real,Zs) )
       => ( pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(aa_lis7252840061267398067l_bool(P,nil_real),nil_nat),nil_real))
         => ( ! [X: real,Xs2: list_real,Y3: nat,Ys5: list_nat,Z3: real,Zs2: list_real] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_nat_nat(size_size_list_nat,Ys5) )
               => ( ( aa_list_nat_nat(size_size_list_nat,Ys5) = aa_list_real_nat(size_size_list_real,Zs2) )
                 => ( pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(aa_lis7252840061267398067l_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(aa_lis7252840061267398067l_bool(P,cons_real(X,Xs2)),aa_list_nat_list_nat(cons_nat(Y3),Ys5)),cons_real(Z3,Zs2))) ) ) )
           => pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(aa_lis7252840061267398067l_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8549_list__induct3,axiom,
    ! [Xs: list_real,Ys: list_nat,Zs: list_bool,P: fun_li4028644538760865171l_bool] :
      ( ( aa_list_real_nat(size_size_list_real,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( ( aa_list_nat_nat(size_size_list_nat,Ys) = aa_list_bool_nat(size_size_list_bool,Zs) )
       => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis4981365976059478484l_bool(P,nil_real),nil_nat),nil_bool))
         => ( ! [X: real,Xs2: list_real,Y3: nat,Ys5: list_nat,Z3: bool,Zs2: list_bool] :
                ( ( aa_list_real_nat(size_size_list_real,Xs2) = aa_list_nat_nat(size_size_list_nat,Ys5) )
               => ( ( aa_list_nat_nat(size_size_list_nat,Ys5) = aa_list_bool_nat(size_size_list_bool,Zs2) )
                 => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis4981365976059478484l_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis4981365976059478484l_bool(P,cons_real(X,Xs2)),aa_list_nat_list_nat(cons_nat(Y3),Ys5)),cons_bool(Z3,Zs2))) ) ) )
           => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis4981365976059478484l_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_8550_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_real,Ws: list_real,P: fun_li1650067531631117579l_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_real_nat(size_size_list_real,Zs) )
       => ( ( aa_list_real_nat(size_size_list_real,Zs) = aa_list_real_nat(size_size_list_real,Ws) )
         => ( pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(aa_lis6354848994543922647l_bool(aa_lis90624776163520116l_bool(P,nil_int),nil_real),nil_real),nil_real))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real,W: real,Ws2: list_real] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_real_nat(size_size_list_real,Zs2) )
                   => ( ( aa_list_real_nat(size_size_list_real,Zs2) = aa_list_real_nat(size_size_list_real,Ws2) )
                     => ( pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(aa_lis6354848994543922647l_bool(aa_lis90624776163520116l_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(aa_lis6354848994543922647l_bool(aa_lis90624776163520116l_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),cons_real(Z3,Zs2)),cons_real(W,Ws2))) ) ) ) )
             => pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(aa_lis6354848994543922647l_bool(aa_lis90624776163520116l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8551_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_real,Ws: list_bool,P: fun_li57212333190706856l_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_real_nat(size_size_list_real,Zs) )
       => ( ( aa_list_real_nat(size_size_list_real,Zs) = aa_list_bool_nat(size_size_list_bool,Ws) )
         => ( pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(aa_lis1658356894974515376l_bool(aa_lis6222097594283734889l_bool(P,nil_int),nil_real),nil_real),nil_bool))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real,W: bool,Ws2: list_bool] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_real_nat(size_size_list_real,Zs2) )
                   => ( ( aa_list_real_nat(size_size_list_real,Zs2) = aa_list_bool_nat(size_size_list_bool,Ws2) )
                     => ( pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(aa_lis1658356894974515376l_bool(aa_lis6222097594283734889l_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(aa_lis1658356894974515376l_bool(aa_lis6222097594283734889l_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),cons_real(Z3,Zs2)),cons_bool(W,Ws2))) ) ) ) )
             => pp(aa_list_bool_bool(aa_lis5910293209478911991l_bool(aa_lis1658356894974515376l_bool(aa_lis6222097594283734889l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8552_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_real,Ws: list_nat,P: fun_li417164691710752487t_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_real_nat(size_size_list_real,Zs) )
       => ( ( aa_list_real_nat(size_size_list_real,Zs) = aa_list_nat_nat(size_size_list_nat,Ws) )
         => ( pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(aa_lis2133206188042513587t_bool(aa_lis3648891015361452112t_bool(P,nil_int),nil_real),nil_real),nil_nat))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real,W: nat,Ws2: list_nat] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_real_nat(size_size_list_real,Zs2) )
                   => ( ( aa_list_real_nat(size_size_list_real,Zs2) = aa_list_nat_nat(size_size_list_nat,Ws2) )
                     => ( pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(aa_lis2133206188042513587t_bool(aa_lis3648891015361452112t_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(aa_lis2133206188042513587t_bool(aa_lis3648891015361452112t_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),cons_real(Z3,Zs2)),aa_list_nat_list_nat(cons_nat(W),Ws2))) ) ) ) )
             => pp(aa_list_nat_bool(aa_lis587257317721978902t_bool(aa_lis2133206188042513587t_bool(aa_lis3648891015361452112t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8553_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_real,Ws: list_int,P: fun_li2812955053305622411t_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_real_nat(size_size_list_real,Zs) )
       => ( ( aa_list_real_nat(size_size_list_real,Zs) = aa_list_int_nat(size_size_list_int,Ws) )
         => ( pp(aa_list_int_bool(aa_lis126974366075332282t_bool(aa_lis4437612097023134039t_bool(aa_lis4335467048402002676t_bool(P,nil_int),nil_real),nil_real),nil_int))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real,W: int,Ws2: list_int] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_real_nat(size_size_list_real,Zs2) )
                   => ( ( aa_list_real_nat(size_size_list_real,Zs2) = aa_list_int_nat(size_size_list_int,Ws2) )
                     => ( pp(aa_list_int_bool(aa_lis126974366075332282t_bool(aa_lis4437612097023134039t_bool(aa_lis4335467048402002676t_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_int_bool(aa_lis126974366075332282t_bool(aa_lis4437612097023134039t_bool(aa_lis4335467048402002676t_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),cons_real(Z3,Zs2)),aa_list_int_list_int(cons_int(W),Ws2))) ) ) ) )
             => pp(aa_list_int_bool(aa_lis126974366075332282t_bool(aa_lis4437612097023134039t_bool(aa_lis4335467048402002676t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8554_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_bool,Ws: list_real,P: fun_li636819950999198860l_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_bool_nat(size_size_list_bool,Zs) )
       => ( ( aa_list_bool_nat(size_size_list_bool,Zs) = aa_list_real_nat(size_size_list_real,Ws) )
         => ( pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(aa_lis1085736148594123412l_bool(aa_lis2409104332336385869l_bool(P,nil_int),nil_real),nil_bool),nil_real))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: bool,Zs2: list_bool,W: real,Ws2: list_real] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_bool_nat(size_size_list_bool,Zs2) )
                   => ( ( aa_list_bool_nat(size_size_list_bool,Zs2) = aa_list_real_nat(size_size_list_real,Ws2) )
                     => ( pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(aa_lis1085736148594123412l_bool(aa_lis2409104332336385869l_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(aa_lis1085736148594123412l_bool(aa_lis2409104332336385869l_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),cons_bool(Z3,Zs2)),cons_real(W,Ws2))) ) ) ) )
             => pp(aa_list_real_bool(aa_lis1167625458524959379l_bool(aa_lis1085736148594123412l_bool(aa_lis2409104332336385869l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8555_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_bool,Ws: list_bool,P: fun_li2826078509111078631l_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_bool_nat(size_size_list_bool,Zs) )
       => ( ( aa_list_bool_nat(size_size_list_bool,Zs) = aa_list_bool_nat(size_size_list_bool,Ws) )
         => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis1730967974872196275l_bool(aa_lis7197491278626754640l_bool(P,nil_int),nil_real),nil_bool),nil_bool))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: bool,Zs2: list_bool,W: bool,Ws2: list_bool] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_bool_nat(size_size_list_bool,Zs2) )
                   => ( ( aa_list_bool_nat(size_size_list_bool,Zs2) = aa_list_bool_nat(size_size_list_bool,Ws2) )
                     => ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis1730967974872196275l_bool(aa_lis7197491278626754640l_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis1730967974872196275l_bool(aa_lis7197491278626754640l_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),cons_bool(Z3,Zs2)),cons_bool(W,Ws2))) ) ) ) )
             => pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aa_lis1730967974872196275l_bool(aa_lis7197491278626754640l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8556_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_bool,Ws: list_nat,P: fun_li1938945883818180328t_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_bool_nat(size_size_list_bool,Zs) )
       => ( ( aa_list_bool_nat(size_size_list_bool,Zs) = aa_list_nat_nat(size_size_list_nat,Ws) )
         => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis9131688112241455088t_bool(aa_lis6065156870740858281t_bool(P,nil_int),nil_real),nil_bool),nil_nat))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: bool,Zs2: list_bool,W: nat,Ws2: list_nat] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_bool_nat(size_size_list_bool,Zs2) )
                   => ( ( aa_list_bool_nat(size_size_list_bool,Zs2) = aa_list_nat_nat(size_size_list_nat,Ws2) )
                     => ( pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis9131688112241455088t_bool(aa_lis6065156870740858281t_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis9131688112241455088t_bool(aa_lis6065156870740858281t_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),cons_bool(Z3,Zs2)),aa_list_nat_list_nat(cons_nat(W),Ws2))) ) ) ) )
             => pp(aa_list_nat_bool(aa_lis7136190398932771567t_bool(aa_lis9131688112241455088t_bool(aa_lis6065156870740858281t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8557_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_bool,Ws: list_int,P: fun_li4334736245413050252t_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_bool_nat(size_size_list_bool,Zs) )
       => ( ( aa_list_bool_nat(size_size_list_bool,Zs) = aa_list_int_nat(size_size_list_int,Ws) )
         => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis2212721984367299732t_bool(aa_lis6751732903781408845t_bool(P,nil_int),nil_real),nil_bool),nil_int))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: bool,Zs2: list_bool,W: int,Ws2: list_int] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_bool_nat(size_size_list_bool,Zs2) )
                   => ( ( aa_list_bool_nat(size_size_list_bool,Zs2) = aa_list_int_nat(size_size_list_int,Ws2) )
                     => ( pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis2212721984367299732t_bool(aa_lis6751732903781408845t_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis2212721984367299732t_bool(aa_lis6751732903781408845t_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),cons_bool(Z3,Zs2)),aa_list_int_list_int(cons_int(W),Ws2))) ) ) ) )
             => pp(aa_list_int_bool(aa_lis6675907447286124947t_bool(aa_lis2212721984367299732t_bool(aa_lis6751732903781408845t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8558_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_nat,Ws: list_real,P: fun_li6575135422763014119l_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_nat_nat(size_size_list_nat,Zs) )
       => ( ( aa_list_nat_nat(size_size_list_nat,Zs) = aa_list_real_nat(size_size_list_real,Ws) )
         => ( pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(aa_lis7252840061267398067l_bool(aa_lis8627625277897029968l_bool(P,nil_int),nil_real),nil_nat),nil_real))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: nat,Zs2: list_nat,W: real,Ws2: list_real] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_nat_nat(size_size_list_nat,Zs2) )
                   => ( ( aa_list_nat_nat(size_size_list_nat,Zs2) = aa_list_real_nat(size_size_list_real,Ws2) )
                     => ( pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(aa_lis7252840061267398067l_bool(aa_lis8627625277897029968l_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(aa_lis7252840061267398067l_bool(aa_lis8627625277897029968l_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),aa_list_nat_list_nat(cons_nat(Z3),Zs2)),cons_real(W,Ws2))) ) ) ) )
             => pp(aa_list_real_bool(aa_lis4500580016340999446l_bool(aa_lis7252840061267398067l_bool(aa_lis8627625277897029968l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8559_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_real,Zs: list_nat,Ws: list_bool,P: fun_li8709814001568230604l_bool] :
      ( ( aa_list_int_nat(size_size_list_int,Xs) = aa_list_real_nat(size_size_list_real,Ys) )
     => ( ( aa_list_real_nat(size_size_list_real,Ys) = aa_list_nat_nat(size_size_list_nat,Zs) )
       => ( ( aa_list_nat_nat(size_size_list_nat,Zs) = aa_list_bool_nat(size_size_list_bool,Ws) )
         => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis4981365976059478484l_bool(aa_lis5964784873068787085l_bool(P,nil_int),nil_real),nil_nat),nil_bool))
           => ( ! [X: int,Xs2: list_int,Y3: real,Ys5: list_real,Z3: nat,Zs2: list_nat,W: bool,Ws2: list_bool] :
                  ( ( aa_list_int_nat(size_size_list_int,Xs2) = aa_list_real_nat(size_size_list_real,Ys5) )
                 => ( ( aa_list_real_nat(size_size_list_real,Ys5) = aa_list_nat_nat(size_size_list_nat,Zs2) )
                   => ( ( aa_list_nat_nat(size_size_list_nat,Zs2) = aa_list_bool_nat(size_size_list_bool,Ws2) )
                     => ( pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis4981365976059478484l_bool(aa_lis5964784873068787085l_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis4981365976059478484l_bool(aa_lis5964784873068787085l_bool(P,aa_list_int_list_int(cons_int(X),Xs2)),cons_real(Y3,Ys5)),aa_list_nat_list_nat(cons_nat(Z3),Zs2)),cons_bool(W,Ws2))) ) ) ) )
             => pp(aa_list_bool_bool(aa_lis5949604092718491675l_bool(aa_lis4981365976059478484l_bool(aa_lis5964784873068787085l_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_8560_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2))) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_mw(A),B)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_8561_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_do,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_acs,N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8562_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_do,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_acq(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8563_ln__inj__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( ( aa_real_real(ln_ln_real,X3) = aa_real_real(ln_ln_real,Y) )
        <=> ( X3 = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_8564_ln__less__cancel__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X3)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_8565_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_nat_set_nat(set_ord_atMost_nat,K)),aa_list_nat_list_nat(cons_nat(aa_nat_nat(suc,K)),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_8566_ln__le__cancel__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X3)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_8567_ln__eq__zero__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( ( aa_real_real(ln_ln_real,X3) = zero_zero_real )
      <=> ( X3 = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_8568_ln__gt__zero__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X3)) ) ) ).

% ln_gt_zero_iff
tff(fact_8569_ln__less__zero__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X3)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),one_one_real)) ) ) ).

% ln_less_zero_iff
tff(fact_8570_ln__ge__zero__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3)) ) ) ).

% ln_ge_zero_iff
tff(fact_8571_ln__le__zero__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X3)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real)) ) ) ).

% ln_le_zero_iff
tff(fact_8572_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_8573_ln__less__self,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X3)),X3)) ) ).

% ln_less_self
tff(fact_8574_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_8575_finite__nat__iff__bounded__le,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),aa_nat_set_nat(set_ord_atMost_nat,K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_8576_ln__bound,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X3)),X3)) ) ).

% ln_bound
tff(fact_8577_ln__gt__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X3))) ) ).

% ln_gt_zero
tff(fact_8578_ln__less__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X3)),zero_zero_real)) ) ) ).

% ln_less_zero
tff(fact_8579_ln__gt__zero__imp__gt__one,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X3)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_8580_ln__ge__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X3))) ) ).

% ln_ge_zero
tff(fact_8581_ln__ge__zero__imp__ge__one,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X3)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_8582_ln__add__one__self__le__self,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X3))),X3)) ) ).

% ln_add_one_self_le_self
tff(fact_8583_ln__mult,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X3)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_8584_ln__eq__minus__one,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( ( aa_real_real(ln_ln_real,X3) = aa_real_real(minus_minus_real(X3),one_one_real) )
       => ( X3 = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_8585_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = 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,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_8586_ln__le__minus__one,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X3)),aa_real_real(minus_minus_real(X3),one_one_real))) ) ).

% ln_le_minus_one
tff(fact_8587_ln__add__one__self__le__self2,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X3))),X3)) ) ).

% ln_add_one_self_le_self2
tff(fact_8588_ln__realpow,axiom,
    ! [X3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(ln_ln_real,aa_nat_real(power_power_real(X3),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,X3)) ) ) ).

% ln_realpow
tff(fact_8589_ln__one__minus__pos__upper__bound,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),X3))),aa_real_real(uminus_uminus_real,X3))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_8590_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X3: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),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_aif(A),X3)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aif(B),X3)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_aih(A),B),X3)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_8591_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_8592_list__encode_Ocases,axiom,
    ! [X3: list_nat] :
      ( ( X3 != nil_nat )
     => ~ ! [X: nat,Xs2: list_nat] : X3 != aa_list_nat_list_nat(cons_nat(X),Xs2) ) ).

% list_encode.cases
tff(fact_8593_real__of__nat__div4,axiom,
    ! [N: nat,X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X3))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X3)))) ).

% real_of_nat_div4
tff(fact_8594_real__of__nat__div2,axiom,
    ! [N: nat,X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X3))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X3))))) ).

% real_of_nat_div2
tff(fact_8595_real__of__nat__div3,axiom,
    ! [N: nat,X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X3))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X3)))),one_one_real)) ).

% real_of_nat_div3
tff(fact_8596_complex__mod__minus__le__complex__mod,axiom,
    ! [X3: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,X3))),aa_complex_real(real_V1022390504157884413omplex,X3))) ).

% complex_mod_minus_le_complex_mod
tff(fact_8597_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A))) ).

% complex_mod_triangle_ineq2
tff(fact_8598_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_8599_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_8600_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_8601_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_8602_zdiv__mono1,axiom,
    ! [A: int,A5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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,A5),B))) ) ) ).

% zdiv_mono1
tff(fact_8603_zdiv__mono2,axiom,
    ! [A: int,B5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(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),B5))) ) ) ) ).

% zdiv_mono2
tff(fact_8604_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_8605_zdiv__mono1__neg,axiom,
    ! [A: int,A5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A5),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_8606_zdiv__mono2__neg,axiom,
    ! [A: int,B5: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B5))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B5)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_8607_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).

% div_int_pos_iff
tff(fact_8608_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))) ) ) ).

% div_positive_int
tff(fact_8609_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_8610_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(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_8611_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_8612_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_8613_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_8614_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_8615_int__div__less__self,axiom,
    ! [X3: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X3),K)),X3)) ) ) ).

% int_div_less_self
tff(fact_8616_ln__div,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(X3),Y)) = aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,X3)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_8617_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_geq
tff(fact_8618_msrevs_I1_J,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),K) ) ) ).

% msrevs(1)
tff(fact_8619_finite__int__iff__bounded__le,axiom,
    ! [S3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(abs_abs_int),S3)),aa_int_set_int(set_ord_atMost_int,K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_8620_verit__less__mono__div__int2,axiom,
    ! [A2: int,B3: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_8621_div__eq__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_8622_ln__diff__le,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,X3)),aa_real_real(ln_ln_real,Y))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X3),Y)),Y))) ) ) ).

% ln_diff_le
tff(fact_8623_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_8624_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_8625_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_8626_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,I3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,I3)) ) ) ) ) ).

% split_zdiv
tff(fact_8627_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_8628_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_8629_int__div__minus__is__minus1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_8630_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_8631_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_8632_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_8633_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_8634_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_8635_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_8636_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_8637_int__div__same__is__1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_8638_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ).

% div_le_dividend
tff(fact_8639_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K))) ) ).

% div_le_mono
tff(fact_8640_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q4) ).

% div_mult2_eq
tff(fact_8641_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_8642_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_8643_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_8644_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_8645_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_8646_div__mult__le,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),A)) ).

% div_mult_le
tff(fact_8647_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).

% div_le_mono2
tff(fact_8648_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_8649_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_8650_td__gal__lt,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),B)) ) ) ).

% td_gal_lt
tff(fact_8651_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q4)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4))) ) ) ).

% div_less_iff_less_mult
tff(fact_8652_zdiv__le__dividend,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),A)) ) ) ).

% zdiv_le_dividend
tff(fact_8653_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_8654_zdiv__mult__self,axiom,
    ! [M: int,A: int,N: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))),M) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),M)),N) ) ) ).

% zdiv_mult_self
tff(fact_8655_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_if
tff(fact_8656_div__nat__eqI,axiom,
    ! [N: nat,Q4: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q4))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_8657_split__div,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I3)),J3) )
               => pp(aa_nat_bool(P,I3)) ) ) ) ) ) ).

% split_div
tff(fact_8658_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_8659_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))))) ) ).

% dividend_less_times_div
tff(fact_8660_td__gal,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C))) ) ) ).

% td_gal
tff(fact_8661_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q4)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_8662_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aii(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_8663_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_8664_split__div_H,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P,zero_zero_nat)) )
        | ? [Q6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q6)),M))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q6))))
            & pp(aa_nat_bool(P,Q6)) ) ) ) ).

% split_div'
tff(fact_8665_power__sub,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),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),N)) = 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),N)) ) ) ) ).

% power_sub
tff(fact_8666_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_8667_sgn__div__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) = aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_8668_list__encode_Oelims,axiom,
    ! [X3: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X3) = Y )
     => ( ( ( X3 = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X: nat,Xs2: list_nat] :
              ( ( X3 = aa_list_nat_list_nat(cons_nat(X),Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_list_nat_nat(nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_8669_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I),J)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_8670_list__encode__eq,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X3) = aa_list_nat_nat(nat_list_encode,Y) )
    <=> ( X3 = Y ) ) ).

% list_encode_eq
tff(fact_8671_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_nat_bool(P,M5)) )
    <=> ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X4)) ) ) ).

% all_nat_less
tff(fact_8672_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
          & pp(aa_nat_bool(P,M5)) )
    <=> ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X4)) ) ) ).

% ex_nat_less
tff(fact_8673_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,L),aa_nat_nat(suc,U)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_8674_atMost__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N) ).

% atMost_atLeast0
tff(fact_8675_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N)) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_8676_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N) = aa_set_nat_set_nat(insert_nat(M),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_8677_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N)) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_8678_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_set_nat(insert_nat(M),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N) ) ) ).

% atLeastAtMost_insertL
tff(fact_8679_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N7),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N7)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_8680_list__encode_Osimps_I1_J,axiom,
    aa_list_nat_nat(nat_list_encode,nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_8681_image__Suc__atMost,axiom,
    ! [N: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_8682_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_8683_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,N)),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_8684_list__encode_Osimps_I2_J,axiom,
    ! [X3: nat,Xs: list_nat] : aa_list_nat_nat(nat_list_encode,aa_list_nat_list_nat(cons_nat(X3),Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),aa_list_nat_nat(nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_8685_list__encode_Opelims,axiom,
    ! [X3: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X3) = Y )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X3))
       => ( ( ( X3 = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
         => ~ ! [X: nat,Xs2: list_nat] :
                ( ( X3 = aa_list_nat_list_nat(cons_nat(X),Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_list_nat_nat(nat_list_encode,Xs2)))) )
                 => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),aa_list_nat_list_nat(cons_nat(X),Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_8686_arctan__add,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,X3)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y)))) ) ) ) ).

% arctan_add
tff(fact_8687_nat__mod__eq_H,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),N) = A ) ) ).

% nat_mod_eq'
tff(fact_8688_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) ) ).

% mod_less
tff(fact_8689_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_8690_arctan__eq__zero__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(arctan,X3) = zero_zero_real )
    <=> ( X3 = zero_zero_real ) ) ).

% arctan_eq_zero_iff
tff(fact_8691_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_8692_arctan__less__zero__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real)) ) ).

% arctan_less_zero_iff
tff(fact_8693_zero__less__arctan__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arctan,X3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3)) ) ).

% zero_less_arctan_iff
tff(fact_8694_zero__le__arctan__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3)) ) ).

% zero_le_arctan_iff
tff(fact_8695_arctan__le__zero__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real)) ) ).

% arctan_le_zero_iff
tff(fact_8696_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8697_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: 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),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_8698_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: 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,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_8699_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: 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),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_8700_Suc__mod__mult__self4,axiom,
    ! [N: 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,N),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_8701_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(aa_set_nat_list_nat(linord2614967742042102400et_nat,aa_nat_set_nat(set_ord_lessThan_nat,K)),aa_list_nat_list_nat(cons_nat(K),nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_8702_mod__Suc__eq,axiom,
    ! [M: nat,N: 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),N))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% mod_Suc_eq
tff(fact_8703_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: 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),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),N) ).

% mod_Suc_Suc_eq
tff(fact_8704_nat__mod__eq,axiom,
    ! [B: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),N) = B ) ) ) ).

% nat_mod_eq
tff(fact_8705_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),M)) ).

% mod_less_eq_dividend
tff(fact_8706_arctan__monotone_H,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X3)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone'
tff(fact_8707_arctan__le__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X3)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ).

% arctan_le_iff
tff(fact_8708_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) != N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ) ).

% mod_Suc
tff(fact_8709_mod__induct,axiom,
    ! [P: fun_nat_bool,N: nat,P3: nat,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P3))
         => ( ! [N2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),P3))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P3))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_8710_nat__mod__lem,axiom,
    ! [N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N))
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N) = B ) ) ) ).

% nat_mod_lem
tff(fact_8711_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_less_divisor
tff(fact_8712_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M2),zero_zero_nat))
     => ( ! [M2: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M2),N2)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_8713_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_8714_lessThan__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N) ).

% lessThan_atLeast0
tff(fact_8715_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,N: 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)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),L))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),N)),L) = N ) ) ) ).

% word_rot_lem
tff(fact_8716_nat__minus__mod,axiom,
    ! [N: 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,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),M))),M) = zero_zero_nat ).

% nat_minus_mod
tff(fact_8717_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_8718_mod__nat__sub,axiom,
    ! [X3: nat,Z: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Y) ) ) ).

% mod_nat_sub
tff(fact_8719_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ) ).

% mod_if
tff(fact_8720_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% mod_geq
tff(fact_8721_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_8722_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% le_mod_geq
tff(fact_8723_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_8724_nat__mod__eq__iff,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
    <=> ? [Q12: nat,Q23: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q12)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q23)) ) ).

% nat_mod_eq_iff
tff(fact_8725_msrevs_I2_J,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) ).

% msrevs(2)
tff(fact_8726_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_8727_finite__nat__iff__bounded,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ? [K3: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),aa_nat_set_nat(set_ord_lessThan_nat,K3))) ) ).

% finite_nat_iff_bounded
tff(fact_8728_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ? [K2: nat] : pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,S3),aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ).

% finite_nat_bounded
tff(fact_8729_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)))
     => ( aa_int_set_int(set_or1266510415728281911st_int(M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = aa_set_int_set_int(insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)),aa_int_set_int(set_or1266510415728281911st_int(M),N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_8730_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( aa_int_set_int(set_or1266510415728281911st_int(I),J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( aa_int_set_int(set_or1266510415728281911st_int(I),J) = aa_set_int_set_int(insert_int(I),aa_int_set_int(set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int)),J)) ) ) ) ).

% simp_from_to
tff(fact_8731_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_le_divisor
tff(fact_8732_div__less__mono,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),N) = zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),N))) ) ) ) ) ).

% div_less_mono
tff(fact_8733_mod__nat__add,axiom,
    ! [X3: nat,Z: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y)),Z))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y)),Z))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y)),Z) ) ) ) ) ) ).

% mod_nat_add
tff(fact_8734_nat__mod__eq__lemma,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3))
       => ? [Q3: nat] : X3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_8735_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q4) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ~ ! [S: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S)) ) ) ).

% mod_eq_nat2E
tff(fact_8736_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q4) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ~ ! [S: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S)) ) ) ).

% mod_eq_nat1E
tff(fact_8737_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q4))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% mod_mult2_eq
tff(fact_8738_div__mod__decomp,axiom,
    ! [A2: nat,N: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N)) ).

% div_mod_decomp
tff(fact_8739_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)) ).

% modulo_nat_def
tff(fact_8740_mod__lemma,axiom,
    ! [C: nat,R2: nat,B: nat,Q4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Q4),C))),R2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mod_lemma
tff(fact_8741_split__mod,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,M)) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I3)),J3) )
               => pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_8742_image__Suc__lessThan,axiom,
    ! [N: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N) ).

% image_Suc_lessThan
tff(fact_8743_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = 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,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_8744_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4))) ) ) ) ).

% aset(7)
tff(fact_8745_aset_I5_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)),T)) ) ) ) ) ).

% aset(5)
tff(fact_8746_aset_I4_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( X5 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4) != T ) ) ) ) ) ).

% aset(4)
tff(fact_8747_aset_I3_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( X5 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4) = T ) ) ) ) ) ).

% aset(3)
tff(fact_8748_bset_I7_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),B3))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X5))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4))) ) ) ) ) ).

% bset(7)
tff(fact_8749_bset_I5_J,axiom,
    ! [D4: int,B3: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B3))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X5),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)),T)) ) ) ) ).

% bset(5)
tff(fact_8750_bset_I4_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),B3))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( X5 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4) != T ) ) ) ) ) ).

% bset(4)
tff(fact_8751_bset_I3_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B3))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( X5 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4) = T ) ) ) ) ) ).

% bset(3)
tff(fact_8752_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X: int,K2: int] :
            ( pp(aa_int_bool(P,X))
          <=> pp(aa_int_bool(P,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] : pp(aa_int_bool(P,X_12))
        <=> ? [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D)))
              & pp(aa_int_bool(P,X4)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_8753_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_8754_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4))) ) ) ) ).

% aset(8)
tff(fact_8755_aset_I6_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)),T)) ) ) ) ) ).

% aset(6)
tff(fact_8756_bset_I8_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B3))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B3))
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X5))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4))) ) ) ) ) ).

% bset(8)
tff(fact_8757_bset_I6_J,axiom,
    ! [D4: int,B3: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa_set_int_bool(member_int(Xa4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B3))
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X5),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)),T)) ) ) ) ).

% bset(6)
tff(fact_8758_cppi,axiom,
    ! [D4: int,P: fun_int_bool,P2: fun_int_bool,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( ? [Z4: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X))
           => ( pp(aa_int_bool(P,X))
            <=> pp(aa_int_bool(P2,X)) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
                 => ! [Xb3: int] :
                      ( pp(aa_set_int_bool(member_int(Xb3),A2))
                     => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
             => ( pp(aa_int_bool(P,X))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4))) ) )
         => ( ! [X: int,K2: int] :
                ( pp(aa_int_bool(P2,X))
              <=> pp(aa_int_bool(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),D4)))) )
           => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
            <=> ( ? [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
                    & pp(aa_int_bool(P2,X4)) )
                | ? [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),A2))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_8759_cpmi,axiom,
    ! [D4: int,P: fun_int_bool,P2: fun_int_bool,B3: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( ? [Z4: int] :
          ! [X: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z4))
           => ( pp(aa_int_bool(P,X))
            <=> pp(aa_int_bool(P2,X)) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( pp(aa_set_int_bool(member_int(Xa2),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
                 => ! [Xb3: int] :
                      ( pp(aa_set_int_bool(member_int(Xb3),B3))
                     => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
             => ( pp(aa_int_bool(P,X))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4))) ) )
         => ( ! [X: int,K2: int] :
                ( pp(aa_int_bool(P2,X))
              <=> pp(aa_int_bool(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),D4)))) )
           => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
            <=> ( ? [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
                    & pp(aa_int_bool(P2,X4)) )
                | ? [X4: int] :
                    ( pp(aa_set_int_bool(member_int(X4),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D4)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),B3))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_8760_verit__le__mono__div,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),N))) ) ) ).

% verit_le_mono_div
tff(fact_8761_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,N)),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_8762_sumr__cos__zero__one,axiom,
    ! [N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aij),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8763_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_nat_real(semiri5074537144036343181t_real,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(X3),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X3)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_8764_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(X3),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X3))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_8765_exp__le__cancel__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X3)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ).

% exp_le_cancel_iff
tff(fact_8766_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_8767_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_8768_exp__eq__one__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(exp_real,X3) = one_one_real )
    <=> ( X3 = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_8769_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8770_one__less__exp__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3)) ) ).

% one_less_exp_iff
tff(fact_8771_exp__less__one__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X3)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real)) ) ).

% exp_less_one_iff
tff(fact_8772_exp__le__one__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X3)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real)) ) ).

% exp_le_one_iff
tff(fact_8773_one__le__exp__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3)) ) ).

% one_le_exp_iff
tff(fact_8774_exp__ln__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X3)) = X3 )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3)) ) ).

% exp_ln_iff
tff(fact_8775_exp__ln,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X3)) = X3 ) ) ).

% exp_ln
tff(fact_8776_Word_Omod__minus__cong,axiom,
    ! [B: int,B5: int,X3: int,X7: int,Y: int,Y7: int,Z6: int] :
      ( ( B = B5 )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X3),B5) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X7),B5) )
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y),B5) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y7),B5) )
         => ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X7),Y7) = Z6 )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Z6),B5) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_8777_mod__plus__cong,axiom,
    ! [B: int,B5: int,X3: int,X7: int,Y: int,Y7: int,Z6: int] :
      ( ( B = B5 )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X3),B5) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X7),B5) )
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y),B5) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Y7),B5) )
         => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X7),Y7) = Z6 )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Z6),B5) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_8778_not__exp__less__zero,axiom,
    ! [X3: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X3)),zero_zero_real)) ).

% not_exp_less_zero
tff(fact_8779_exp__gt__zero,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(exp_real,X3))) ).

% exp_gt_zero
tff(fact_8780_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X: real] : aa_real_real(exp_real,X) = Y ) ).

% exp_total
tff(fact_8781_exp__ge__zero,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X3))) ).

% exp_ge_zero
tff(fact_8782_not__exp__le__zero,axiom,
    ! [X3: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X3)),zero_zero_real)) ).

% not_exp_le_zero
tff(fact_8783_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_8784_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).

% neg_mod_bound
tff(fact_8785_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_8786_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_8787_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_8788_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
     => ? [Q3: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_8789_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 )
    <=> ? [Q6: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q6) ) ).

% zmod_eq_0_iff
tff(fact_8790_exp__gt__one,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X3))) ) ).

% exp_gt_one
tff(fact_8791_exp__ge__add__one__self,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X3)),aa_real_real(exp_real,X3))) ).

% exp_ge_add_one_self
tff(fact_8792_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ) ) ).

% neg_mod_conj
tff(fact_8793_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ) ) ).

% pos_mod_conj
tff(fact_8794_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_8795_int__mod__ge,axiom,
    ! [A: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),N))) ) ) ).

% int_mod_ge
tff(fact_8796_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_8797_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_8798_int__mod__lem,axiom,
    ! [N: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),N)) )
      <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),N) = B ) ) ) ).

% int_mod_lem
tff(fact_8799_int__mod__eq,axiom,
    ! [B: int,N: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),N))
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),N) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),N) )
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),N) = B ) ) ) ) ).

% int_mod_eq
tff(fact_8800_int__mod__le_H,axiom,
    ! [B: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),N)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),N)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),N))) ) ).

% int_mod_le'
tff(fact_8801_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = zero_zero_int ) )
      & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = 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_8802_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = zero_zero_int ) )
      & ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = 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_8803_nonneg__mod__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ) ).

% nonneg_mod_div
tff(fact_8804_zdiv__mono__strict,axiom,
    ! [A2: int,B3: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),N) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),N) = zero_zero_int )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_8805_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_8806_exp__ge__add__one__self__aux,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X3)),aa_real_real(exp_real,X3))) ) ).

% exp_ge_add_one_self_aux
tff(fact_8807_lemma__exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y))
     => ? [X: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(minus_minus_real(Y),one_one_real)))
          & ( aa_real_real(exp_real,X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_8808_ln__ge__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X3)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y)),X3)) ) ) ).

% ln_ge_iff
tff(fact_8809_ln__x__over__x__mono,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X3)),X3))) ) ) ).

% ln_x_over_x_mono
tff(fact_8810_int__mod__ge_H,axiom,
    ! [B: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),N))) ) ) ).

% int_mod_ge'
tff(fact_8811_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_8812_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_8813_mod__power__lem,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),M)) = zero_zero_int ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),M)) = aa_nat_int(power_power_int(A),N) ) ) ) ) ).

% mod_power_lem
tff(fact_8814_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_8815_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_8816_split__zmod,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,N)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_8817_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_8818_zmod__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_8819_mod__add__if__z,axiom,
    ! [X3: int,Z: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
             => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),Z))
                 => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y) ) )
                & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),Z))
                 => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),Z) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y)),Z) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_8820_mod__sub__if__z,axiom,
    ! [X3: int,Z: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
             => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
                 => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)),Z) = aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y) ) )
                & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X3))
                 => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)),Z) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_8821_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = 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)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_8822_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = 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(modulo_modulo_int,A),B) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_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_8823_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = 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(modulo_modulo_int,A),B) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,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_8824_finite__int__iff__bounded,axiom,
    ! [S3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
    <=> ? [K3: int] : pp(aa_set_int_bool(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(abs_abs_int),S3)),aa_int_set_int(set_ord_lessThan_int,K3))) ) ).

% finite_int_iff_bounded
tff(fact_8825_verit__le__mono__div__int,axiom,
    ! [A2: int,B3: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)),aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),N)),zero_zero_int)),one_one_int),zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_8826_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_8827_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_8828_Maclaurin__exp__lt,axiom,
    ! [X3: real,N: nat] :
      ( ( X3 != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X3)))
            & ( aa_real_real(exp_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aik(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N))),aa_nat_real(power_power_real(X3),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8829_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),N))
     => ( take_nat(M,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_8830_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_8831_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_8832_length__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_nat(size_size_list_nat,upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_8833_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K)) )
     => ( append_nat(upt(I,J),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_8834_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_8835_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_8836_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J))
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_8837_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_list_nat_nat(groups4561878855575611511st_nat,upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dl),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N)) ) ) ).

% sum_list_upt
tff(fact_8838_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(suc,upt(M,N)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_8839_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_te,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_8840_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
      ( ( aa_list_nat_list_nat(cons_nat(M),aa_list_nat_list_nat(cons_nat(N),Ns)) = upt(M,Q4) )
    <=> ( aa_list_nat_list_nat(cons_nat(N),Ns) = upt(aa_nat_nat(suc,M),Q4) ) ) ).

% upt_conv_Cons_Cons
tff(fact_8841_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_8842_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_te,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% map_add_upt
tff(fact_8843_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M,N))) ).

% sorted_wrt_upt
tff(fact_8844_sorted__upt,axiom,
    ! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N))) ).

% sorted_upt
tff(fact_8845_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( upt(I,J) = append_nat(Xs,Ys) )
      <=> ? [K3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),J))
            & ( upt(I,K3) = Xs )
            & ( upt(K3,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_8846_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( upt(I,J) = aa_list_nat_list_nat(cons_nat(I),upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_8847_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_ail,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_8848_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( append_nat(upt(zero_zero_nat,I),upt(I,J)) = upt(zero_zero_nat,J) ) ) ).

% upt_append
tff(fact_8849_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M) = aa_list_nat_set_nat(set_nat2,upt(N,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_8850_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_8851_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = aa_list_nat_set_nat(set_nat2,upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_8852_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X3: nat,Xs: list_nat] :
      ( ( upt(I,J) = aa_list_nat_list_nat(cons_nat(X3),Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( I = X3 )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_8853_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = aa_list_nat_set_nat(set_nat2,upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_8854_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = aa_list_nat_list_nat(cons_nat(I),upt(aa_nat_nat(suc,I),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_8855_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( upt(L,H) = append_nat(Is1,aa_list_nat_list_nat(cons_nat(I),Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa_nat_nat(suc,I),H) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),H)) ) ) ).

% upt_eq_lel_conv
tff(fact_8856_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),aa_list_nat_list_nat(cons_nat(J),nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_8857_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),aa_list_nat_list_nat(cons_nat(J),nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_8858_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B9: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_aim(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B9),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_8859_Maclaurin__exp__le,axiom,
    ! [X3: real,N: nat] :
    ? [T6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X3)))
      & ( aa_real_real(exp_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aik(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N))),aa_nat_real(power_power_real(X3),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_8860_tl__upt,axiom,
    ! [M: nat,N: nat] : tl_nat(upt(M,N)) = upt(aa_nat_nat(suc,M),N) ).

% tl_upt
tff(fact_8861_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% fact_mono_nat
tff(fact_8862_fact__ge__self,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_self
tff(fact_8863_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_8864_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_8865_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_8866_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_8867_upt__filter__extend,axiom,
    ! [U: nat,U2: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,U),U2))
     => ( ! [I2: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,U),I2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),U2)) )
           => ~ pp(aa_nat_bool(P,I2)) )
       => ( filter_nat(P,upt(zero_zero_nat,U)) = filter_nat(P,upt(zero_zero_nat,U2)) ) ) ) ).

% upt_filter_extend
tff(fact_8868_binomial__Suc__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),N) = aa_nat_nat(suc,N) ).

% binomial_Suc_n
tff(fact_8869_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_8870_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_8871_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% binomial_eq_0_iff
tff(fact_8872_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_8873_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8874_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% zero_less_binomial_iff
tff(fact_8875_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8876_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ).

% Suc_times_binomial
tff(fact_8877_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_8878_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_8879_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),M)) ).

% choose_mult_lemma
tff(fact_8880_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% binomial_le_pow
tff(fact_8881_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ).

% zero_less_binomial
tff(fact_8882_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_8883_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),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,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8884_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_8885_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_8886_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ain,M)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_8887_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_8888_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_8889_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aio,R2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),N))),N) ).

% sum_choose_lower
tff(fact_8890_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aip,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_8891_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aip,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8892_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_8893_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8894_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_8895_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiq(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_8896_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_air(M),N),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R2) ).

% vandermonde
tff(fact_8897_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_8898_binomial,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ais(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_8899_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_8900_cos__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,N)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% cos_coeff_Suc
tff(fact_8901_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( aa_list_nat_nat(hd_nat,upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_8902_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8903_sin__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,N)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% sin_coeff_Suc
tff(fact_8904_smod__int__range,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),aa_int_set_int(set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,B))),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,B)),one_one_int)))) ) ).

% smod_int_range
tff(fact_8905_merge__pure__and,axiom,
    ! [A: bool,B: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_bool_assn(pure_assn,A)),aa_bool_assn(pure_assn,B)) = aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,A),B)) ).

% merge_pure_and
tff(fact_8906_and__extract__pure__right__ctx__iff,axiom,
    ! [P: assn,Q: assn,B: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),aa_bool_assn(pure_assn,B))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q)),aa_bool_assn(pure_assn,B)) ).

% and_extract_pure_right_ctx_iff
tff(fact_8907_and__extract__pure__left__ctx__iff,axiom,
    ! [P: assn,B: bool,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_bool_assn(pure_assn,B))),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q)),aa_bool_assn(pure_assn,B)) ).

% and_extract_pure_left_ctx_iff
tff(fact_8908_smod__int__0__mod,axiom,
    ! [X3: int] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,zero_zero_int),X3) = zero_zero_int ).

% smod_int_0_mod
tff(fact_8909_smod__int__mod__0,axiom,
    ! [X3: int] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,X3),zero_zero_int) = X3 ).

% smod_int_mod_0
tff(fact_8910_and__extract__pure__right__iff,axiom,
    ! [P: assn,B: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),aa_bool_assn(pure_assn,B)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,one_one_assn),P)),aa_bool_assn(pure_assn,B)) ).

% and_extract_pure_right_iff
tff(fact_8911_and__extract__pure__left__iff,axiom,
    ! [B: bool,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_bool_assn(pure_assn,B)),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,one_one_assn),Q)),aa_bool_assn(pure_assn,B)) ).

% and_extract_pure_left_iff
tff(fact_8912_ent__conjE2,axiom,
    ! [B3: assn,C2: assn,A2: assn] :
      ( entails(B3,C2)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B3),C2) ) ).

% ent_conjE2
tff(fact_8913_ent__conjE1,axiom,
    ! [A2: assn,C2: assn,B3: assn] :
      ( entails(A2,C2)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B3),C2) ) ).

% ent_conjE1
tff(fact_8914_ent__conjI,axiom,
    ! [A2: assn,B3: assn,C2: assn] :
      ( entails(A2,B3)
     => ( entails(A2,C2)
       => entails(A2,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B3),C2)) ) ) ).

% ent_conjI
tff(fact_8915_norm__assertion__simps_I9_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X3) = bot_bot_assn ).

% norm_assertion_simps(9)
tff(fact_8916_norm__assertion__simps_I10_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),bot_bot_assn) = bot_bot_assn ).

% norm_assertion_simps(10)
tff(fact_8917_smod__int__compares_I1_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),B)) ) ) ).

% smod_int_compares(1)
tff(fact_8918_smod__int__compares_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B))) ) ) ).

% smod_int_compares(2)
tff(fact_8919_smod__int__compares_I4_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),zero_zero_int)) ) ) ).

% smod_int_compares(4)
tff(fact_8920_smod__int__compares_I6_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B))) ) ) ).

% smod_int_compares(6)
tff(fact_8921_smod__int__compares_I7_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),zero_zero_int)) ) ) ).

% smod_int_compares(7)
tff(fact_8922_smod__int__compares_I8_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B))) ) ) ).

% smod_int_compares(8)
tff(fact_8923_smod__mod__positive,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% smod_mod_positive
tff(fact_8924_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_8925_prod__Suc__fact,axiom,
    ! [N: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_8926_smod__int__compares_I3_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B))) ) ) ).

% smod_int_compares(3)
tff(fact_8927_smod__int__compares_I5_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A),B)),aa_int_int(uminus_uminus_int,B))) ) ) ).

% smod_int_compares(5)
tff(fact_8928_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_dl),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,N)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8929_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_dl),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_8930_upto_Opelims,axiom,
    ! [X3: int,Xa: int,Y: list_int] :
      ( ( upto(X3,Xa) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Xa)))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
               => ( Y = aa_list_int_list_int(cons_int(X3),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int),Xa)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
               => ( Y = nil_int ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Xa))) ) ) ) ).

% upto.pelims
tff(fact_8931_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(I),J)))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = aa_list_int_list_int(cons_int(I),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_8932_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J))
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_8933_less__eq__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% less_eq_assn_def
tff(fact_8934_minus__assn__def,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),uminus_uminus_assn(B)) ).

% minus_assn_def
tff(fact_8935_assn__aci_I4_J,axiom,
    ! [X3: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Y) ).

% assn_aci(4)
tff(fact_8936_assn__aci_I3_J,axiom,
    ! [X3: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Z)) ).

% assn_aci(3)
tff(fact_8937_assn__aci_I1_J,axiom,
    ! [X3: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X3) ).

% assn_aci(1)
tff(fact_8938_norm__assertion__simps_I31_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),X3) = X3 ).

% norm_assertion_simps(31)
tff(fact_8939_norm__assertion__simps_I14_J,axiom,
    ! [X3: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% norm_assertion_simps(14)
tff(fact_8940_sorted__upto,axiom,
    ! [M: int,N: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),upto(M,N))) ).

% sorted_upto
tff(fact_8941_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I),J)) = aa_set_int_int(groups1705073143266064639nt_int(aTP_Lamm_dm),aa_int_set_int(set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_8942_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = aa_list_int_list_int(cons_int(I),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_8943_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = aa_list_int_list_int(cons_int(I),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_8944_upto_Oelims,axiom,
    ! [X3: int,Xa: int,Y: list_int] :
      ( ( upto(X3,Xa) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
         => ( Y = aa_list_int_list_int(cons_int(X3),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int),Xa)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_8945_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_8946_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_8947_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),aa_list_int_list_int(cons_int(J),nil_int)) ) ) ).

% upto_rec2
tff(fact_8948_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = aa_set_int_int(groups1705073143266064639nt_int(aTP_Lamm_dm),aa_int_set_int(set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_8949_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),aa_list_int_list_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_8950_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_list_int_list_int(cons_int(aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(2)
tff(fact_8951_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_list_int_list_int(cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(3)
tff(fact_8952_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_list_int_list_int(cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(4)
tff(fact_8953_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = aa_list_int_list_int(cons_int(aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(1)
tff(fact_8954_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_8955_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: 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)),N))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_8956_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_8957_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_num_real(numeral_numeral_real,W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W2))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_8958_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N)),aa_nat_real(semiri5074537144036343181t_real,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_8959_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = aa_list_nat_list_nat(cons_nat(aa_num_nat(numeral_numeral_nat,M)),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_8960_real__of__int__div4,axiom,
    ! [N: int,X3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X3))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X3)))) ).

% real_of_int_div4
tff(fact_8961_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real))) ) ).

% int_le_real_less
tff(fact_8962_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)),aa_int_real(ring_1_of_int_real,M))) ) ).

% int_less_real_le
tff(fact_8963_real__of__int__div2,axiom,
    ! [N: int,X3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,N)),aa_int_real(ring_1_of_int_real,X3))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X3))))) ).

% real_of_int_div2
tff(fact_8964_real__of__int__div3,axiom,
    ! [N: int,X3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X3))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X3)))),one_one_real)) ).

% real_of_int_div3
tff(fact_8965_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M),numera1916890842035813515d_enat(N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(1)
tff(fact_8966_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M),numera1916890842035813515d_enat(N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(2)
tff(fact_8967_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),Ma))
          & ? [M2: nat] :
              ( ( aa_nat_option_nat(some_nat,M2) = aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary) )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ) ) ) ).

% mintlistlength
tff(fact_8968_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_8969_i0__less,axiom,
    ! [N: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,N)
    <=> ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_8970_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_8971_two__realpow__ge__two,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ) ).

% two_realpow_ge_two
tff(fact_8972_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X3))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% member_bound
tff(fact_8973_semiring__norm_I83_J,axiom,
    ! [N: num] : one != aa_num_num(bit0,N) ).

% semiring_norm(83)
tff(fact_8974_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_8975_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_nat_option_nat(some_nat,M) = aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,T) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% misiz
tff(fact_8976_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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))),N))) ) ).

% two_powr_height_bound_deg
tff(fact_8977_insert__simp__mima,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X3 = Mi )
        | ( X3 = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% insert_simp_mima
tff(fact_8978_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(T),X3) = aa_nat_option_nat(some_nat,Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% helpyd
tff(fact_8979_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(T),X3) = aa_nat_option_nat(some_nat,Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).

% helpypredd
tff(fact_8980_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X3))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Y)),X3)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_8981_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X3)),X3)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_8982_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X3)),Y))
           => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y))
              | ( X3 = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_8983_count__buildup,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)))) ).

% count_buildup
tff(fact_8984_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(6)
tff(fact_8985_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% semiring_norm(13)
tff(fact_8986_semiring__norm_I11_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M),one) = M ).

% semiring_norm(11)
tff(fact_8987_semiring__norm_I12_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),N) = N ).

% semiring_norm(12)
tff(fact_8988_delt__out__of__range,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% delt_out_of_range
tff(fact_8989_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(78)
tff(fact_8990_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(71)
tff(fact_8991_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),one)) ).

% semiring_norm(75)
tff(fact_8992_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,one),N)) ).

% semiring_norm(68)
tff(fact_8993_del__single__cont,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X3 = Mi )
        & ( X3 = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) ) ) ) ).

% del_single_cont
tff(fact_8994_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N)) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).

% set_n_deg_not_0
tff(fact_8995_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList),Summary)),X3)),one_one_nat)) ) ).

% tdeletemimi'
tff(fact_8996_count__buildup_H,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),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))),N))))) ).

% count_buildup'
tff(fact_8997_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_8998_succ__min,axiom,
    ! [Deg: nat,X3: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_nat_option_nat(some_nat,Mi) ) ) ) ).

% succ_min
tff(fact_8999_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X3: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3))
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_nat_option_nat(some_nat,Ma) ) ) ) ).

% pred_max
tff(fact_9000_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)),one_one_real)))) ) ).

% cnt_bound'
tff(fact_9001_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D))),L) ).

% bit_concat_def
tff(fact_9002_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X3: nat] : pp(aa_nat_bool(time_TBOUND_bool(vEBT_V854960066525838166emberi(T,Ti,X3)),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_9003_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_VEBT_VEBT_set_nat(vEBT_VEBT_set_vebt,T)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_9004_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_9005_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),one)) ).

% semiring_norm(69)
tff(fact_9006_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit0,N))) ).

% semiring_norm(76)
tff(fact_9007_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% Suc_numeral
tff(fact_9008_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_9009_add__2__eq__Suc_H,axiom,
    ! [N: nat] : 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_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_9010_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_9011_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_9012_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_9013_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_9014_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_9015_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% half_nonnegative_int_iff
tff(fact_9016_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% half_negative_int_iff
tff(fact_9017_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_9018_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X3: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,Y)
     => ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X3),aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(Y),Z)) = aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X3),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_9019_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ord_le2932123472753598470d_enat(N,zero_z5237406670263579293d_enat)
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_9020_i0__lb,axiom,
    ! [N: extended_enat] : ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat,N) ).

% i0_lb
tff(fact_9021_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N))
    <=> ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,M)
        & ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,N) ) ) ).

% enat_0_less_mult_iff
tff(fact_9022_not__iless0,axiom,
    ! [N: extended_enat] : ~ ord_le72135733267957522d_enat(N,zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_9023_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_9024_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_9025_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% pos2
tff(fact_9026_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: 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))),N) ).

% Suc_double_not_eq_double
tff(fact_9027_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% double_not_eq_Suc_double
tff(fact_9028_n__less__equal__power__2,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% n_less_equal_power_2
tff(fact_9029_le__num__One__iff,axiom,
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),one))
    <=> ( X3 = one ) ) ).

% le_num_One_iff
tff(fact_9030_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M))) ) ).

% self_le_ge2_pow
tff(fact_9031_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_9032_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_9033_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_9034_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_9035_nat__induct2,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_9036_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))) ) ).

% diff_le_diff_pow
tff(fact_9037_realpow__square__minus__le,axiom,
    ! [U: real,X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% realpow_square_minus_le
tff(fact_9038_pos__mod__sign2,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% pos_mod_sign2
tff(fact_9039_nmod2,axiom,
    ! [N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int )
      | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% nmod2
tff(fact_9040_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_9041_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),zero_zero_int)) ).

% not_exp_less_eq_0_int
tff(fact_9042_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% binomial_maximum
tff(fact_9043_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_antimono
tff(fact_9044_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N))) ).

% two_realpow_ge_one
tff(fact_9045_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N))) ).

% binomial_maximum'
tff(fact_9046_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_mono
tff(fact_9047_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% binomial_le_pow2
tff(fact_9048_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_9049_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_9050_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_9051_two__pow__div__gt__le,axiom,
    ! [V: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,V),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% two_pow_div_gt_le
tff(fact_9052_nat__add__offset__less,axiom,
    ! [Y: nat,N: nat,X3: nat,M: nat,Sz: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),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_9053_nat__power__less__diff,axiom,
    ! [N: nat,Q4: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Q4),aa_nat_nat(aa_nat_fun_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),N)))) ) ).

% nat_power_less_diff
tff(fact_9054_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9055_nat__le__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_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))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,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)),N)),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_9056_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_9057_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).

% binomial_strict_mono
tff(fact_9058_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_9059_axxmod2,axiom,
    ! [X3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),X3)),X3)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int )
      & ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),X3)),X3)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% axxmod2
tff(fact_9060_axxdiv2,axiom,
    ! [X3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),X3)),X3)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = X3 )
      & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),X3)),X3)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = X3 ) ) ).

% axxdiv2
tff(fact_9061_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% square_fact_le_2_fact
tff(fact_9062_exp__half__le2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9063_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_9064_choose__row__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% choose_row_sum
tff(fact_9065_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_9066_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_9067_less__two__pow__divD,axiom,
    ! [X3: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),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))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% less_two_pow_divD
tff(fact_9068_less__two__pow__divI,axiom,
    ! [X3: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),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))),N)),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_9069_nat__less__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_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)),N)),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_9070_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_9071_int__bit__induct,axiom,
    ! [P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(P,zero_zero_int))
     => ( pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,one_one_int)))
       => ( ! [K2: int] :
              ( pp(aa_int_bool(P,K2))
             => ( ( K2 != zero_zero_int )
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
         => ( ! [K2: int] :
                ( pp(aa_int_bool(P,K2))
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => pp(aa_int_bool(P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_9072_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_9073_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_9074_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_9075_choose__square__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ait,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N) ).

% choose_square_sum
tff(fact_9076_nat__div__eq__Suc__0__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(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_9077_power__2__mult__step__le,axiom,
    ! [N3: nat,N: nat,K6: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)),K6)),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))),N)),K)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K6),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))),N)),K))) ) ) ).

% power_2_mult_step_le
tff(fact_9078_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_9079_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_9080_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_9081_exp__bound,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X3)),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% exp_bound
tff(fact_9082_sb__dec__lem_H,axiom,
    ! [K: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_9083_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N))) ).

% mask_eq_sum_exp_nat
tff(fact_9084_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dl),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_9085_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(aa_nat1818925681583257441et_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_9086_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dl),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9087_sb__inc__lem,axiom,
    ! [A: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(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_9088_sb__inc__lem_H,axiom,
    ! [A: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(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_9089_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_9090_sb__dec__lem,axiom,
    ! [K: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(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_9091_ln__one__plus__pos__lower__bound,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(X3),aa_nat_real(power_power_real(X3),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),X3)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_9092_real__exp__bound__lemma,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X3)),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))),X3)))) ) ) ).

% real_exp_bound_lemma
tff(fact_9093_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)))
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2))))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_9094_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiu(A),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_9095_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dl),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_9096_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),N) ).

% Suc_nat_number_of_add
tff(fact_9097_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
     => ( aa_set_int_int(groups4538972089207619220nt_int(aTP_Lamm_dm),aa_int_set_int(set_or1266510415728281911st_int(M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_9098_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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))),M)) ).

% binomial_r_part_sum
tff(fact_9099_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiv,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_9100_exp__lower__Taylor__quadratic,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X3)),aa_real_real(divide_divide_real(aa_nat_real(power_power_real(X3),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,X3))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_9101_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),X3))),X3))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_9102_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(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 = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_13))
               => ( ! [X: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
                     => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_13)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_9103_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)))
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_9104_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_9105_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(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,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_13))
               => ( ! [X: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
                     => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_13)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_9106_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_9107_ln__one__minus__pos__lower__bound,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X3),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),X3)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_9108_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X3))),X3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_9109_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),X3))),X3))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_9110_Tb__T__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,N))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% Tb_T_vebt_buildupi
tff(fact_9111_Tb__T__vebt__buildupi_H,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(vEBT_V9176841429113362141ildupi,N)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% Tb_T_vebt_buildupi'
tff(fact_9112_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(vEBT_VEBT_Tb2,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Tb_T_vebt_buildupi''
tff(fact_9113_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_9114_max__enat__simps_I3_J,axiom,
    ! [Q4: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),Q4) = Q4 ).

% max_enat_simps(3)
tff(fact_9115_max__enat__simps_I2_J,axiom,
    ! [Q4: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(Q4),zero_z5237406670263579293d_enat) = Q4 ).

% max_enat_simps(2)
tff(fact_9116_idiff__0,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat),N) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_9117_idiff__0__right,axiom,
    ! [N: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N),zero_z5237406670263579293d_enat) = N ).

% idiff_0_right
tff(fact_9118_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_9119_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_9120_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_9121_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_9122_div__half__nat,axiom,
    ! [Y: nat,X3: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Y)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Y)) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),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(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))),Y)))),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))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),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(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))),Y))),Y))),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))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),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(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))),Y)))) ) ) ).

% div_half_nat
tff(fact_9123_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X3: nat] : pp(aa_nat_bool(time_htt_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_memberi(Ti,X3),aa_nat_fun_bool_assn(aa_VEB5211276637198020136l_assn(aTP_Lamm_au(T),Ti),X3)),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_9124_vebt__buildupi__rule,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)),vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),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))),N)))) ).

% vebt_buildupi_rule
tff(fact_9125_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_9126_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,N)),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_9127_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit1,M) != aa_num_num(bit0,N) ).

% semiring_norm(89)
tff(fact_9128_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit0,M) != aa_num_num(bit1,N) ).

% semiring_norm(88)
tff(fact_9129_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_9130_semiring__norm_I84_J,axiom,
    ! [N: num] : one != aa_num_num(bit1,N) ).

% semiring_norm(84)
tff(fact_9131_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList),Summary)),X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% tdeletemimi
tff(fact_9132_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(80)
tff(fact_9133_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(73)
tff(fact_9134_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),X3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),X3)),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_9135_Tb_H__cnt,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_VEBT_Tb2,N)),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,N))))) ).

% Tb'_cnt
tff(fact_9136_T__vebt__buildupi__cnt_H,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V441764108873111860ildupi,N))),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,N))))) ).

% T_vebt_buildupi_cnt'
tff(fact_9137_TBOUND__buildupi,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_vebt_buildupi(N)),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))),N)))) ) ).

% TBOUND_buildupi
tff(fact_9138_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),X3)),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_9139_cnt__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)),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_9140_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(7)
tff(fact_9141_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(9)
tff(fact_9142_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X3: nat] : pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(T,Ti,X3)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(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_9143_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),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_9144_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),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_9145_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_9146_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),aa_num_num(bit1,N))) ).

% semiring_norm(14)
tff(fact_9147_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(72)
tff(fact_9148_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(81)
tff(fact_9149_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),one)) ).

% semiring_norm(70)
tff(fact_9150_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit1,N))) ).

% semiring_norm(77)
tff(fact_9151_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X3: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_inserti(Ti,X3),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X3))),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_9152_semiring__norm_I3_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,N)) = aa_num_num(bit1,N) ).

% semiring_norm(3)
tff(fact_9153_semiring__norm_I4_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% semiring_norm(4)
tff(fact_9154_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_9155_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one)) ).

% semiring_norm(8)
tff(fact_9156_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),one)) ).

% semiring_norm(10)
tff(fact_9157_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)))) ).

% semiring_norm(16)
tff(fact_9158_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(74)
tff(fact_9159_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(79)
tff(fact_9160_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_9161_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: 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,N)))) = 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))),N)) ).

% div_Suc_eq_div_add3
tff(fact_9162_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_9163_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: 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,N)))) = 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))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_9164_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_9165_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_9166_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_9167_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_9168_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9169_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_9170_mint__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9171_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_9172_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: 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)))),N) = 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)),N) ).

% Suc_div_eq_add3_div
tff(fact_9173_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: 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)))),N) = 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)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_9174_exp__le,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% exp_le
tff(fact_9175_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_9176_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_9177_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_9178_small__powers__of__2,axiom,
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),X3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),one_one_nat)))) ) ).

% small_powers_of_2
tff(fact_9179_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,T),X3)),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_9180_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,T),X3)),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_9181_space__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_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_9182_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N)),one_one_real)))) ) ).

% space_2_pow_bound
tff(fact_9183_space__cnt,axiom,
    ! [T: vEBT_VEBT] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9184_space__space_H,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))) ).

% space_space'
tff(fact_9185_space_H__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_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_9186_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_tg(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(U),L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_9187_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U))) ).

% finite_atLeastZeroLessThan_integer
tff(fact_9188_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_bool_VEBT_VEBT(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_9189_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.space.simps(1)
tff(fact_9190_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_9191_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,X3) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( 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] :
              ( ( X3 = 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_9192_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_9193_VEBT__internal_Ospace_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,X3) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3 = 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_9194_t__build__cnt,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8646137997579335489_i_l_d,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))) ).

% t_build_cnt
tff(fact_9195_t__buildup__cnt,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,N))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))) ).

% t_buildup_cnt
tff(fact_9196_vebt__buildup__bound,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N)),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_9197_buildup__build__time,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,N))) ).

% buildup_build_time
tff(fact_9198_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_9199_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_9200_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_9201_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_9202_VEBT__internal_Ospace_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),X3))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = 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)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_9203_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,X3) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel),X3))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = 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)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_9204_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1)))
     => ( ! [K2: int,L2: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K2),L2)))
           => ( ( ~ ( pp(aa_set_int_bool(member_int(K2),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(L2),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))) )
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,K2),L2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_9205_setceilmax,axiom,
    ! [S2: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,S2),M))
     => ( ! [X: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Listy)))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N)) )
       => ( ( M = aa_nat_nat(suc,N) )
         => ( ! [X: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(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,N))) ) )
           => ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,S2)) = 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(S2),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_9206_pred__list__to__short,axiom,
    ! [Deg: nat,X3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_9207_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X3: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_9208_high__def,axiom,
    ! [X3: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% high_def
tff(fact_9209_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ).

% high_bound_aux
tff(fact_9210_high__inv,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X3)),N) = Y ) ) ).

% high_inv
tff(fact_9211_log__ceil__idem,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3)) = 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,X3)))) ) ) ).

% log_ceil_idem
tff(fact_9212_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( 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,N))) ) ) ).

% heigt_uplog_rel
tff(fact_9213_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,T),X3))),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_9214_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Deg) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),Deg))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9215_log__one,axiom,
    ! [A: real] : aa_real_real(log(A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_9216_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),X3))),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_9217_zero__less__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(log(A),X3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X3)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_9218_log__less__zero__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(A),X3)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),one_one_real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_9219_one__less__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(log(A),X3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_9220_log__less__one__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(A),X3)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),A)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_9221_log__less__cancel__iff,axiom,
    ! [A: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(A),X3)),aa_real_real(log(A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_9222_log__eq__one,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9223_zero__le__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(log(A),X3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_9224_log__le__zero__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(A),X3)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_9225_one__le__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(log(A),X3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_9226_log__le__one__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(A),X3)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),A)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_9227_log__le__cancel__iff,axiom,
    ! [A: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(A),X3)),aa_real_real(log(A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_9228_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_nat_real(power_power_real(A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_9229_log__base__change,axiom,
    ! [A: real,B: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(log(B),X3) = aa_real_real(divide_divide_real(aa_real_real(log(A),X3)),aa_real_real(log(A),B)) ) ) ) ).

% log_base_change
tff(fact_9230_log__mult,axiom,
    ! [A: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(log(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(A),X3)),aa_real_real(log(A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_9231_le__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(B),M))) ) ) ).

% le_log_of_power
tff(fact_9232_log__divide,axiom,
    ! [A: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(log(A),aa_real_real(divide_divide_real(X3),Y)) = aa_real_real(minus_minus_real(aa_real_real(log(A),X3)),aa_real_real(log(A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_9233_log__base__pow,axiom,
    ! [A: real,N: nat,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(log(aa_nat_real(power_power_real(A),N)),X3) = aa_real_real(divide_divide_real(aa_real_real(log(A),X3)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log_base_pow
tff(fact_9234_log__nat__power,axiom,
    ! [X3: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(log(B),aa_nat_real(power_power_real(X3),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(B),X3)) ) ) ).

% log_nat_power
tff(fact_9235_log__of__power__less,axiom,
    ! [M: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_9236_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( ( B != one_one_real )
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
             => ( aa_real_real(log(A),X3) = 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),X3)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_9237_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X3: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_9238_log__of__power__le,axiom,
    ! [M: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_9239_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_9240_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% le_log2_of_power
tff(fact_9241_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_9242_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,T),X3))),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_9243_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,T),X3))),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_9244_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_9245_log__base__10__eq2,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X3) = 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,X3)) ) ) ).

% log_base_10_eq2
tff(fact_9246_log__base__10__eq1,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X3) = 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,X3)) ) ) ).

% log_base_10_eq1
tff(fact_9247_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,T),X3))),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_9248_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,T),X3))),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_9249_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_9250_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_9251_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N)),one_one_int) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_9252_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(time_htt_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_memberi(Ti,X3),aa_nat_fun_bool_assn(aa_VEB5211276637198020136l_assn(aTP_Lamm_au(T),Ti),X3)),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,N)))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_9253_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_inserti(Ti,X3),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),X3))),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,N)))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_9254_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( ( N = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),Mi))
         => ( ( Ma != Mi )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_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_9255_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg))
     => ( ( Mi != Ma )
       => ( the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_9256_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9257_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_9258_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_9259_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_9260_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% zless_nat_conj
tff(fact_9261_nat__zminus__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_9262_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = zero_zero_int ) ) ) ).

% int_nat_eq
tff(fact_9263_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% zero_less_nat_eq
tff(fact_9264_length__upto,axiom,
    ! [I: int,J: int] : aa_list_int_nat(size_size_list_int,upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_9265_nat__ceiling__le__eq,axiom,
    ! [X3: real,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X3))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_nat_real(semiri5074537144036343181t_real,A))) ) ).

% nat_ceiling_le_eq
tff(fact_9266_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% one_less_nat_eq
tff(fact_9267_numeral__power__less__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),N)),A)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9268_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9269_numeral__power__le__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X3)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),N)),A)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9270_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,X3)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X3)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9271_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_9272_nat__mono,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X3)),aa_int_nat(nat2,Y))) ) ).

% nat_mono
tff(fact_9273_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z6))
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z6) )
        <=> ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_9274_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_12: nat] : pp(aa_nat_bool(P,X_12))
    <=> ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
         => pp(aa_nat_bool(P,aa_int_nat(nat2,X4))) ) ) ).

% all_nat
tff(fact_9275_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
    <=> ? [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
          & pp(aa_nat_bool(P,aa_int_nat(nat2,X4))) ) ) ).

% ex_nat
tff(fact_9276_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% nat_mono_iff
tff(fact_9277_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_9278_nat__le__iff,axiom,
    ! [X3: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X3)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_9279_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_9280_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ) ).

% int_eq_iff
tff(fact_9281_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_9282_real__nat__ceiling__ge,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X3))))) ).

% real_nat_ceiling_ge
tff(fact_9283_nat__plus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_plus_as_int
tff(fact_9284_nat__times__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_times_as_int
tff(fact_9285_nat__minus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_minus_as_int
tff(fact_9286_nat__div__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_div_as_int
tff(fact_9287_nat__mod__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_mod_as_int
tff(fact_9288_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_9289_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa_int_nat(nat2,W2) )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_9290_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_9291_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N6: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N6) )
           => pp(aa_nat_bool(P,N6)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_9292_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W2))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_9293_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z6))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_add_distrib
tff(fact_9294_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ) ).

% le_nat_iff
tff(fact_9295_Suc__as__int,axiom,
    ! [X5: nat] : aa_nat_nat(suc,X5) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),one_one_int)) ).

% Suc_as_int
tff(fact_9296_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ).

% nat_mult_distrib
tff(fact_9297_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z6))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z6),Z))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_diff_distrib
tff(fact_9298_nat__diff__distrib_H,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X3)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9299_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_9300_nat__div__distrib,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X3),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X3)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_9301_nat__div__distrib_H,axiom,
    ! [Y: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X3),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X3)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_9302_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_9303_nat__mod__distrib,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X3),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X3)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_9304_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_9305_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_9306_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_9307_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z6))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9308_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( 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)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( 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)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ) ).

% nat_abs_int_diff
tff(fact_9309_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
     => ( aa_int_set_int(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_9310_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z6),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z6),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6))) ) ) ) ).

% diff_nat_eq_if
tff(fact_9311_pred__less__length__list,axiom,
    ! [Deg: nat,X3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3),aa_nat_option_nat(some_nat,Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% pred_less_length_list
tff(fact_9312_pred__lesseq__max,axiom,
    ! [Deg: nat,X3: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma))
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3),aa_nat_option_nat(some_nat,Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% pred_lesseq_max
tff(fact_9313_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X3))
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% succ_greatereq_min
tff(fact_9314_bit__split__inv,axiom,
    ! [X3: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),D)),D) = X3 ).

% bit_split_inv
tff(fact_9315_low__def,axiom,
    ! [X3: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% low_def
tff(fact_9316_low__inv,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X3)),N) = X3 ) ) ).

% low_inv
tff(fact_9317_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),X3)) ) ) ) ).

% both_member_options_ding
tff(fact_9318_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
          | ( X3 = Mi )
          | ( X3 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_9319_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
        & ( ( X3 = Mi )
          | ( X3 = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)))
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_9320_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X3: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_9321_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X3: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X3 != Ma )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_9322_insert__simp__norm,axiom,
    ! [X3: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X3 != Ma )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Ma))),Deg,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_9323_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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),H)),L) )
               => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Newlist = aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),Newnode) )
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_9324_del__x__mi__lets__in__not__minNull,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X3 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
                   => ( ( Newlist = aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),Newnode) )
                     => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_9325_del__x__not__mia,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L)),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))))))),Ma))),Deg,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),H))))),Ma))),Deg,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),Summary)) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_9326_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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),H)),L) )
               => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Sn = aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H) )
                   => ( ( Newlist = aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),Newnode) )
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,Sn)),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,Sn)))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,Sn)))))))),Ma))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_9327_del__x__not__mi,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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),H)),L) )
               => ( ( Newlist = aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),Newnode) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)))
                   => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))))))),Ma))),Deg,Newlist),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H)) ) )
                      & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_9328_del__x__mia,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X3 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = 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BT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))),Ma))),Deg,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,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) ) ) ) ).

% del_x_mia
tff(fact_9329_del__x__mi__lets__in__minNull,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X3 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
                   => ( ( Newlist = aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),Newnode) )
                     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( ( Sn = aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H) )
                         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,Sn)),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,Sn)))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,Sn)))))))),Ma))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_9330_del__x__mi__lets__in,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X3 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L) )
                   => ( ( Newlist = aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),Newnode) )
                     => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))))))),Ma))),Deg,Newlist),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H)) ) )
                        & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_9331_del__x__mi,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X3 = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L)),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))))))))),Ma))),Deg,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),H))))),Ma))),Deg,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,H),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H)),L))),Summary)) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_9332_del__in__range,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = 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um_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),Ma))),Deg,aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),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_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)) ) ) ) ) ).

% del_in_range
tff(fact_9333_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X3: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X3) = if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_9334_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X3: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_9335_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList),S2)),X3))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_9336_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList),Vd)),X3))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_9337_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3))
    <=> ( ( X3 != Mi )
       => ( ( X3 != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_9338_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList),Vc)),X3))
    <=> ( ( X3 = Mi )
        | ( X3 = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_9339_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X3),Xa))
      <=> pp(Y) )
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B2) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S)
               => ( pp(Y)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_9340_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X3),Xa))
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => pp(B2) )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_9341_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X3),Xa))
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ( ( ( Xa = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => pp(B2) )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_9342_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X3),Xa))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2)
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_9343_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(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 = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                   => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),X_12))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
                       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_13)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_9344_vebt__member_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X3),Xa))
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => pp(B2) )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
             => ~ ( ( Xa != Mi2 )
                 => ( ( Xa != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_9345_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X3),Xa))
     => ( ! [Uu2: bool,Uv2: bool] : X3 != aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2)
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_9346_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X3),Xa))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_9347_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Mi)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Ma)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_9348_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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))),one_one_nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_9349_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Ma)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),X3),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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_VEBT_VEBT(vEBT_vebt_insert(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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)))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_9350_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(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,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                   => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),X_12))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList)))
                       => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X),X_13)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_9351_vebt__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X3),Xa))
      <=> pp(Y) )
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B2) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => pp(Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2)
             => pp(Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
               => pp(Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_9352_vebt__member_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X3),Xa))
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ( ( ( Xa = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => pp(B2) )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_9353_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,X3),Xa) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),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] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Ma2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_9354_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,X3),Xa) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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))),one_one_nat) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_9355_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,A1),A22))
     => ( ( ? [A3: bool,B2: bool] : A1 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X5),N2)) )
                 => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M2))
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                     => ( ( M2 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                         => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2)))
                                 => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X5),N2)) )
                   => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M2))
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                       => ( ( M2 = aa_nat_nat(suc,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                           => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_1))
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2)))
                                   => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X5),N2)) )
                     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M2))
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                         => ( ( M2 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                             => ( ! [I4: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                                   => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),X_12))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2)))
                                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I4 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                & ! [X5: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N2) = I4 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N2))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X5))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X5),N2)) )
                       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M2))
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                           => ( ( M2 = aa_nat_nat(suc,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2) )
                               => ( ! [I4: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                                     => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),X_12))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2)))
                                         => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_1)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I4 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                  & ! [X5: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N2) = I4 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N2))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X5))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_9356_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,A1),A22))
    <=> ( ( ? [A4: bool,B4: bool] : A1 = aa_bool_VEBT_VEBT(vEBT_Leaf(A4),B4)
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList4: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList4),Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N6)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N6))
            & ( 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))),N6) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),N6) )
            & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_12))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4)))
               => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
        | ? [TreeList4: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList4),Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N6)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N6)))
            & ( 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,N6)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),aa_nat_nat(suc,N6)) )
            & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_12))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4)))
               => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
        | ? [TreeList4: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList4),Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N6)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N6))
            & ( 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))),N6) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),N6) )
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)))
               => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),X_12))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4)))
                 => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N6) = I3 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N6))) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N6) = I3 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N6))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X4))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma3)) ) ) ) ) ) )
        | ? [TreeList4: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList4),Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4)))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X4),N6)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N6)))
            & ( 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,N6)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),aa_nat_nat(suc,N6)) )
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6))))
               => ( ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),X_12))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4)))
                 => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N6) = I3 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N6))) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N6) = I3 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N6))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X4))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_9357_vebt__insert_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(X3),Xa) = Y )
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),B2) ) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),fTrue) ) )
                    & ( ( Xa != one_one_nat )
                     => ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S) )
             => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S) )
               => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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_VEBT_VEBT(vEBT_vebt_insert(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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)))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_9358_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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))),one_one_nat) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_9359_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X3: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3))
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3))
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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))),one_one_nat) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_9360_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,
    ! [X3: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3)) )
       => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = one_one_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3)) )
       => ( ( ( ( X3 = Mi )
              & ( X3 = Ma ) )
           => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = one_one_nat ) )
          & ( ~ ( ( X3 = Mi )
                & ( X3 = Ma ) )
           => ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),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(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),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_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),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))),one_one_nat) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_9361_vebt__succ_Osimps_I6_J,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_option_nat(some_nat,Mi) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_9362_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X3: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3))
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_option_nat(some_nat,Ma) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3))
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_option_nat(vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3),aa_nat_option_nat(some_nat,Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_9363_vebt__delete_Osimps_I7_J,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3)) )
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi))
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3)) )
       => ( ( ( ( X3 = Mi )
              & ( X3 = Ma ) )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) )
          & ( ~ ( ( X3 = Mi )
                & ( X3 = Ma ) )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),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(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_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nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),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))))))))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),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_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary))))))),X3)),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))))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_9364_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,X3),Xa) = Y )
     => ( ( ? [Uu2: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2)
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)
           => ( ? [N2: nat] : Xa = aa_nat_nat(suc,N2)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( Y = one_one_nat ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_9365_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,X3),Xa) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: bool,Uw2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)
           => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
             => ( ? [Va: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( Y = one_one_nat ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_9366_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),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(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),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(aa_nat_option_nat(vEBT_vebt_succ(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_9367_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3)),one_one_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,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),aa_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(aa_nat_option_nat(vEBT_vebt_pred(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),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>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_9368_vebt__delete_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(X3),Xa) = Y )
     => ( ! [A3: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
           => ( ( Xa = zero_zero_nat )
             => ( Y != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),B2) ) ) )
       => ( ! [A3: bool] :
              ( ? [B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
             => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != aa_bool_VEBT_VEBT(vEBT_Leaf(A3),fFalse) ) ) )
         => ( ! [A3: bool,B2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
               => ( ? [N2: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,N2))
                 => ( Y != aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X3 = 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] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                   => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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(aa_Pro4773402355715522498at_nat(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] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) )
                            & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Ma2))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fu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2860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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))))))))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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))))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_9369_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,X3),Xa) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
           => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
             => ( ? [N2: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,N2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X3 = 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] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y = one_one_nat ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_9370_vebt__succ_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_succ(X3),Xa) = Y )
     => ( ! [Uu2: bool,B2: bool] :
            ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2) )
           => ( ( Xa = zero_zero_nat )
             => ~ ( ( pp(B2)
                   => ( Y = aa_nat_option_nat(some_nat,one_one_nat) ) )
                  & ( ~ pp(B2)
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)
           => ( ? [N2: nat] : Xa = aa_nat_nat(suc,N2)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( Y = aa_nat_option_nat(some_nat,Mi2) ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_option_nat(vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_9371_vebt__pred_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_pred(X3),Xa) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
         => ( ( Xa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: bool] :
              ( ? [Uw2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)
             => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
               => ~ ( ( pp(A3)
                     => ( Y = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
                    & ( ~ pp(A3)
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A3: bool,B2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
               => ( ? [Va: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va))
                 => ~ ( ( pp(B2)
                       => ( Y = aa_nat_option_nat(some_nat,one_one_nat) ) )
                      & ( ~ pp(B2)
                       => ( ( pp(A3)
                           => ( Y = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
                          & ( ~ pp(A3)
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( Y = aa_nat_option_nat(some_nat,Ma2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                             => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_option_nat(vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa),aa_nat_option_nat(some_nat,Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_9372_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,X3),Xa) = Y )
     => ( ( ? [Uu2: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2)
         => ( ( Xa = zero_zero_nat )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)
           => ( ? [N2: nat] : Xa = aa_nat_nat(suc,N2)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),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(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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(aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))))),one_one_nat)))) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_9373_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,X3),Xa) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: bool,Uw2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)
           => ( ( Xa = 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: bool,B2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
               => ( ? [Va: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va))
                 => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),one_one_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,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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(aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))))),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_9374_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = 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).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_9375_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,X3),Xa) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
           => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
             => ( ? [N2: nat] : Xa = aa_nat_nat(suc,aa_nat_nat(suc,N2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X3 = 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] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != 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ide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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)))))),one_one_nat)))))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_9376_in__children__def,axiom,
    ! [N: nat,TreeList: list_VEBT_VEBT,X3: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList),X3))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N))) ) ).

% in_children_def
tff(fact_9377_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),zero_zero_nat))) ) ) )
         => ( ! [A3: bool,B2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,zero_zero_nat)))) ) ) )
           => ( ! [A3: bool,B2: bool] :
                  ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
                 => ! [N2: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,aa_nat_nat(suc,N2))))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)),Xa))) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2)),Xa))) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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 )
                         => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2)),Xa))) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = 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ide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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)))))),one_one_nat)))))) )
                           => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_9378_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [Uu2: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2)),zero_zero_nat))) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2) )
               => ! [N2: nat] :
                    ( ( Xa = aa_nat_nat(suc,N2) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)),aa_nat_nat(suc,N2)))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Xa))) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2)),Xa))) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Xa))) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),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(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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(aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))))),one_one_nat)))) )
                         => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_9379_vebt__succ_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_succ(X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [Uu2: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( ( pp(B2)
                     => ( Y = aa_nat_option_nat(some_nat,one_one_nat) ) )
                    & ( ~ pp(B2)
                     => ( Y = none_nat ) ) )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2)),zero_zero_nat))) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2) )
               => ! [N2: nat] :
                    ( ( Xa = aa_nat_nat(suc,N2) )
                   => ( ( Y = none_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)),aa_nat_nat(suc,N2)))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = none_nat )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Xa))) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y = none_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2)),Xa))) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = none_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Xa))) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                             => ( Y = aa_nat_option_nat(some_nat,Mi2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                             => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_option_nat(vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_nat_option_nat(vEBT_vebt_succ(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                         => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_9380_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)),zero_zero_nat))) ) ) )
         => ( ! [A3: bool,Uw2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2) )
               => ( ( Xa = 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) )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)),aa_nat_nat(suc,zero_zero_nat)))) ) ) )
           => ( ! [A3: bool,B2: bool] :
                  ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
                 => ! [Va: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va))))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)),Xa))) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2)),Xa))) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Xa))) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),one_one_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,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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(aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))))),one_one_nat)))) )
                           => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_9381_vebt__pred_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_pred(X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)),zero_zero_nat))) ) ) )
         => ( ! [A3: bool,Uw2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( ( pp(A3)
                       => ( Y = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
                      & ( ~ pp(A3)
                       => ( Y = none_nat ) ) )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)),aa_nat_nat(suc,zero_zero_nat)))) ) ) )
           => ( ! [A3: bool,B2: bool] :
                  ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
                 => ! [Va: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( ( pp(B2)
                           => ( Y = aa_nat_option_nat(some_nat,one_one_nat) ) )
                          & ( ~ pp(B2)
                           => ( ( pp(A3)
                               => ( Y = aa_nat_option_nat(some_nat,zero_zero_nat) ) )
                              & ( ~ pp(A3)
                               => ( Y = none_nat ) ) ) ) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va))))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3) )
                   => ( ( Y = none_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)),Xa))) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = none_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2)),Xa))) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = none_nat )
                         => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Xa))) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( Y = aa_nat_option_nat(some_nat,Ma2) ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_option_nat(vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa),aa_nat_option_nat(some_nat,Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(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)))))),aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_nat_option_nat(vEBT_vebt_pred(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                           => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_9382_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),zero_zero_nat))) ) ) )
         => ( ! [A3: bool,B2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,zero_zero_nat)))) ) ) )
           => ( ! [A3: bool,B2: bool] :
                  ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
                 => ! [N2: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,aa_nat_nat(suc,N2))))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)),Xa))) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2)),Xa))) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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 )
                         => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2)),Xa))) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y = one_one_nat ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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))),one_one_nat) ) ) ) ) )
                           => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_9383_vebt__delete_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),B2) )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),zero_zero_nat))) ) ) )
         => ( ! [A3: bool,B2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),fFalse) )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,zero_zero_nat)))) ) ) )
           => ( ! [A3: bool,B2: bool] :
                  ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
                 => ! [N2: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,aa_nat_nat(suc,N2))))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)),Xa))) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                     => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2)),Xa))) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                         => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2)),Xa))) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) )
                              & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y = 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2860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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))))))))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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(aa_VEB7679222860772546456on_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(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Summary2))))))),Xa)),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))))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)) ) ) ) ) )
                           => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_9384_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [Uu2: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),B2)),zero_zero_nat))) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2) )
               => ! [N2: nat] :
                    ( ( Xa = aa_nat_nat(suc,N2) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uv2),Uw2)),aa_nat_nat(suc,N2)))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Xa))) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vc2),Vd2)),Xa))) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Xa))) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                             => ( Y = one_one_nat ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                             => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) )
                         => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_9385_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)),zero_zero_nat))) ) ) )
         => ( ! [A3: bool,Uw2: bool] :
                ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2) )
               => ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),Uw2)),aa_nat_nat(suc,zero_zero_nat)))) ) ) )
           => ( ! [A3: bool,B2: bool] :
                  ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
                 => ! [Va: nat] :
                      ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va))))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)),Xa))) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Vd2),Ve2)),Xa))) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Xa))) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( Y = one_one_nat ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) )
                           => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_9386_vebt__insert_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( ( ( Xa = zero_zero_nat )
                   => ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(fTrue),B2) ) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),fTrue) ) )
                      & ( ( Xa != one_one_nat )
                       => ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) ) ) ) ) )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa))) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S) )
               => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S) )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S)),Xa))) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S) )
                 => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S) )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S)),Xa))) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2)),Xa))) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Xa),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),aa_VEB5888937730619358530T_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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_VEBT_VEBT(vEBT_vebt_insert(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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)))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_9387_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = one_one_nat )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa))) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S) )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S)),Xa))) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S)),Xa))) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2)),Xa))) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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))),one_one_nat) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_9388_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = one_one_nat )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xa))) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2)),Xa))) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xa))) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Ma2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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)))))) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_9389_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,X3),Xa) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2)
             => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),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] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),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))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_9390_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,X3),Xa) = Y )
     => ( ( ? [A3: bool,B2: bool] : X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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,S: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),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] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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))),one_one_nat)) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_9391_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,S2: vEBT_VEBT,X3: 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),S2)),X3) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_9392_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,S2: vEBT_VEBT,X3: 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),S2)),X3) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_9393_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: bool,B: bool,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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_9394_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),X3) = 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_9395_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: 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,V)),TreeList),Summary)),X3) = 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_9396_insersimp,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_13))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9397_insertsimp,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9398_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),zero_zero_nat,Uy),Uz)),X3) = 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_9399_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),X3) = 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_9400_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: bool,B: bool,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_bool_VEBT_VEBT(vEBT_Leaf(A),B)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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_9401_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),X3)),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_9402_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,T),X3)),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_9403_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),X3))),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_9404_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X3: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,T),X3))),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_9405_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Ma)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X3)),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))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),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_9406_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,X3: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X3) = 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)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Mi)),Mi),X3)),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))),one_one_nat)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_9407_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa))) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S) )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S)),Xa))) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S)),Xa))) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2)),Xa))) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),Mi2),Xa)),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))),one_one_nat)) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_9408_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,
    ! [X3: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,X3),Xa) = Y )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xa))) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2)),Xa))) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xa))) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(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))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa),Ma2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),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))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_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),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)))))))))) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_9409_vebt__member_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X3),Xa))
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa)))
               => ( ( ( Xa = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B2) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xa))) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2) )
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2)),Xa))) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xa))) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa)))
                       => ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_9410_vebt__member_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X3),Xa))
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa)))
               => ~ ( ( ( Xa = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B2) )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
               => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa)))
                 => ~ ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_9411_vebt__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X3),Xa))
      <=> pp(Y) )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( pp(Y)
                <=> ( ( ( Xa = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B2) )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ~ pp(Y)
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xa))) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ~ pp(Y)
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),zero_zero_nat,Uy2),Uz2)),Xa))) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ~ pp(Y)
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xa))) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)),Xa))) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_9412_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X3),Xa))
      <=> pp(Y) )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( ( pp(Y)
                <=> ( ( ( Xa = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B2) )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa))) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ( ~ pp(Y)
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)),Xa))) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S)),Xa))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_9413_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X3),Xa))
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa)))
               => ~ ( ( ( Xa = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => pp(B2) )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S) )
               => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S)),Xa)))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_9414_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X3),Xa))
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [A3: bool,B2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2) )
             => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(A3),B2)),Xa)))
               => ( ( ( Xa = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => pp(B2) )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)),Xa))) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S) )
                 => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S)),Xa)))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_9415_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X3),Xa))
      <=> pp(Y) )
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ( ~ pp(Y)
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)),Xa))) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ( ~ pp(Y)
                 => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Xa))) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2)),Xa))) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
                     => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)),Xa))) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                       => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)),Xa))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_9416_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X3),Xa))
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X3 = aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2) )
             => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_bool_VEBT_VEBT(vEBT_Leaf(Uu2),Uv2)),Xa))) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ~ pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Xa))) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
                 => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2)),Xa)))
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)),Xa)))
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)),Xa)))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_9417_lowi__h,axiom,
    ! [X3: nat,N: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_lowi(X3,N),aa_nat_fun_nat_assn(aTP_Lamm_aiw(X3),N)) ).

% lowi_h
tff(fact_9418_TBOUND__lowi,axiom,
    ! [X3: nat,N: nat] : pp(aa_nat_bool(time_TBOUND_nat(vEBT_VEBT_lowi(X3,N)),one_one_nat)) ).

% TBOUND_lowi
tff(fact_9419_lowi__hT,axiom,
    ! [X3: nat,N: nat] : pp(aa_nat_bool(time_htt_nat(one_one_assn,vEBT_VEBT_lowi(X3,N),aa_nat_fun_nat_assn(aTP_Lamm_aiw(X3),N)),one_one_nat)) ).

% lowi_hT
tff(fact_9420_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X3),Xa))
     => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(X3),Xa)))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
             => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2)),Xa)))
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
               => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)),Xa)))
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X3 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                 => ( pp(aa_Pro1009365227942981478t_bool(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)),Xa)))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,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))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_9421_highi__h,axiom,
    ! [X3: nat,N: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_highi(X3,N),aa_nat_fun_nat_assn(aTP_Lamm_aix(X3),N)) ).

% highi_h
tff(fact_9422_highi__hT,axiom,
    ! [X3: nat,N: nat] : pp(aa_nat_bool(time_htt_nat(one_one_assn,vEBT_VEBT_highi(X3,N),aa_nat_fun_nat_assn(aTP_Lamm_aix(X3),N)),one_one_nat)) ).

% highi_hT
tff(fact_9423_monoseq__arctan__series,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => topolo6980174941875973593q_real(aTP_Lamm_aiy(X3)) ) ).

% monoseq_arctan_series
tff(fact_9424_TBOUND__highi,axiom,
    ! [X3: nat,N: nat] : pp(aa_nat_bool(time_TBOUND_nat(vEBT_VEBT_highi(X3,N)),one_one_nat)) ).

% TBOUND_highi
tff(fact_9425_monoseq__realpow,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => topolo6980174941875973593q_real(power_power_real(X3)) ) ) ).

% monoseq_realpow
tff(fact_9426_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A2))
       => ( aa_set_nat_nat(nat_set_encode,aa_set_nat_set_nat(insert_nat(N),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_9427_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_9428_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% set_bit_negative_int_iff
tff(fact_9429_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_9430_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se7879613467334960850it_int(N),K))) ).

% set_bit_greater_eq
tff(fact_9431_set__encode__eq,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B3))
       => ( ( aa_set_nat_nat(nat_set_encode,A2) = aa_set_nat_nat(nat_set_encode,B3) )
        <=> ( A2 = B3 ) ) ) ) ).

% set_encode_eq
tff(fact_9432_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_9433_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_9434_sin__coeff__def,axiom,
    ! [X5: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5))
       => ( aa_nat_real(sin_coeff,X5) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5))
       => ( aa_nat_real(sin_coeff,X5) = aa_real_real(divide_divide_real(aa_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,X5),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X5)) ) ) ) ).

% sin_coeff_def
tff(fact_9435_ln__series,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X3) = suminf_real(aTP_Lamm_aiz(X3)) ) ) ) ).

% ln_series
tff(fact_9436_arctan__series,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => ( aa_real_real(arctan,X3) = suminf_real(aTP_Lamm_aja(X3)) ) ) ).

% arctan_series
tff(fact_9437_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_9438_dvd__1__left,axiom,
    ! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).

% dvd_1_left
tff(fact_9439_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_9440_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( ( K = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_9441_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ajb,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9442_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc
tff(fact_9443_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_9444_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),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)))) ) ) ).

% odd_Suc_div_two
tff(fact_9445_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),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))) ) ) ).

% even_Suc_div_two
tff(fact_9446_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_9447_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ).

% even_diff_nat
tff(fact_9448_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_9449_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_9450_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% dvd_diff_nat
tff(fact_9451_gcd__nat_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% gcd_nat.extremum
tff(fact_9452_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_9453_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_9454_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat))
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_9455_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_9456_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).

% dvd_pos_nat
tff(fact_9457_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_9458_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_minus_self
tff(fact_9459_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).

% dvd_diffD
tff(fact_9460_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_9461_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_9462_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X3: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y3)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_9463_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(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) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_9464_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = D2 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_9465_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% dvd_imp_le
tff(fact_9466_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_9467_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_9468_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9469_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_9470_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q4) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_9471_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% dvd_fact
tff(fact_9472_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ajb,M)))) ) ).

% finite_divisors_nat
tff(fact_9473_div2__even__ext__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y)) )
       => ( X3 = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_9474_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_9475_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_9476_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_9477_dvd__minus__add,axiom,
    ! [Q4: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q4),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q4)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q4)))) ) ) ) ).

% dvd_minus_add
tff(fact_9478_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_9479_diff__mod__le,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),D))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9480_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_9481_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_9482_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),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(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_9483_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).

% even_set_encode_iff
tff(fact_9484_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% even_mod_4_div_2
tff(fact_9485_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% odd_mod_4_div_2
tff(fact_9486_Bernoulli__inequality__even,axiom,
    ! [N: nat,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X3))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X3)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_9487_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,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))))))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,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(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.simps(3)
tff(fact_9488_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_9489_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ajc(F),G)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ajd(F)),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aje(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_9490_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,X3) = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N2: nat] :
                ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                     => ( Y = 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,N2),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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                     => ( Y = 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,N2),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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_9491_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,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))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,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(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'.simps(3)
tff(fact_9492_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,X3) = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [N2: nat] :
                ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                     => ( 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(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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                     => ( 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(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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_9493_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,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))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,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(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.simps(3)
tff(fact_9494_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,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))))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,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(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'.simps(3)
tff(fact_9495_cos__coeff__def,axiom,
    ! [X5: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5))
       => ( aa_nat_real(cos_coeff,X5) = aa_real_real(divide_divide_real(aa_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,X5),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X5)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X5))
       => ( aa_nat_real(cos_coeff,X5) = zero_zero_real ) ) ) ).

% cos_coeff_def
tff(fact_9496_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] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_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(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,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(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))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,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)))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = 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,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(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,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(vEBT_V8346862874174094_d_u_p,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))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_9497_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] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_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,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,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(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))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = 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,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(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,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(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_9498_vebt__buildup_Oelims,axiom,
    ! [X3: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,X3) = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) ) )
       => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) ) )
         => ~ ! [Va: nat] :
                ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_9499_VEBT__internal_OTb_Oelims,axiom,
    ! [X3: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,X3) = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
       => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
         => ~ ! [N2: nat] :
                ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                     => ( Y = 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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                     => ( Y = 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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_9500_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,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,X3) = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [Va: nat] :
                ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_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(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,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(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))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,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)))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( 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(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),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,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(vEBT_V8346862874174094_d_u_p,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))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_9501_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,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,X3) = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ( ( ( X3 = 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] :
                ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_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,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,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(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))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( 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(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),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,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(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_9502_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X3: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,X3) = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != one_one_int ) )
         => ~ ! [N2: nat] :
                ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                     => ( Y = 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,N2),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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                     => ( Y = 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,N2),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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_9503_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X3: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,X3) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),X3))
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),zero_zero_nat)) ) )
         => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = one_one_int )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                       => ( Y = 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,N2),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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                       => ( Y = 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,N2),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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_9504_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,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,X3) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),X3))
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat)) ) )
         => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_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,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,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(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))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( 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(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),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,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(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) )
                   => ~ pp(aa_nat_bool(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_9505_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,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,X3) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),X3))
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat)) ) )
         => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_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(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,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(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))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,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)))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( 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(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),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,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(vEBT_V8346862874174094_d_u_p,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))) ) ) )
                   => ~ pp(aa_nat_bool(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_9506_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( aa_set_int_int(lattic8263393255366662781ax_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ajf,N))) = aa_int_int(abs_abs_int,N) ) ) ).

% Max_divisors_self_int
tff(fact_9507_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R2))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9508_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9509_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9510_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9511_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_9512_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ) ).

% zdvd_not_zless
tff(fact_9513_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),T))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),T))) ) ) ).

% zdvd_mono
tff(fact_9514_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),N)))
     => ( ( K != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_9515_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ajf,I)))) ) ).

% finite_divisors_int
tff(fact_9516_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_9517_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I))) ) ) ).

% dvd_imp_le_int
tff(fact_9518_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
       => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9519_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( ( L = zero_zero_int )
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).

% mod_int_pos_iff
tff(fact_9520_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)),M))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_9521_int__div__sub__1,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),M))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int)),M) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),M)),one_one_int) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int)),M) = aa_int_int(aa_int_fun_int_int(divide_divide_int,N),M) ) ) ) ) ).

% int_div_sub_1
tff(fact_9522_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z)),M))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_9523_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ) ).

% even_nat_iff
tff(fact_9524_emep1,axiom,
    ! [N: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(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,N),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,N),D)),one_one_int) ) ) ) ) ).

% emep1
tff(fact_9525_eme1p,axiom,
    ! [N: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(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),N)),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,N),D)) ) ) ) ) ).

% eme1p
tff(fact_9526_vebt__buildup_Opelims,axiom,
    ! [X3: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,X3) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X3))
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_bool_VEBT_VEBT(vEBT_Leaf(fFalse),fFalse) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_9527_VEBT__internal_OTb_Opelims,axiom,
    ! [X3: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,X3) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),X3))
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),zero_zero_nat)) ) )
         => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                       => ( Y = 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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                       => ( Y = 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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_9528_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,X3) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),X3))
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),zero_zero_nat)) ) )
         => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                       => ( 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(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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                       => ( 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(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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_9529_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X3: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,X3) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),X3))
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat)) ) )
         => ( ( ( X3 = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X3 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                       => ( Y = 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,N2),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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
                       => ( Y = 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,N2),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,N2),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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_9530_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_ajg) ).

% pi_series
tff(fact_9531_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_9532_pi__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).

% pi_gt_zero
tff(fact_9533_pi__not__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),zero_zero_real)) ).

% pi_not_less_zero
tff(fact_9534_pi__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

% pi_ge_zero
tff(fact_9535_pi__ge__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) ).

% pi_ge_two
tff(fact_9536_pi__half__neq__zero,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_9537_pi__half__le__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9538_pi__half__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9539_pi__half__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9540_minus__pi__half__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(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_9541_arctan__inverse,axiom,
    ! [X3: real] :
      ( ( X3 != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),X3)) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X3)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,X3)) ) ) ).

% arctan_inverse
tff(fact_9542_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),zero_zero_real))
            & ( aa_real_real(cos_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ajh(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X3),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_9543_Maclaurin__cos__expansion2,axiom,
    ! [X3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X3))
            & ( aa_real_real(cos_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ajh(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X3),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_9544_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),X3))
            & ( aa_real_real(sin_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aji(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X3),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_9545_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_9546_sin__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_9547_sin__npi2,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = zero_zero_real ).

% sin_npi2
tff(fact_9548_sin__npi__int,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_npi_int
tff(fact_9549_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_9550_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_9551_sin__2npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_9552_sin__int__2pin,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_int_2pin
tff(fact_9553_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_9554_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_9555_sin__cos__npi,axiom,
    ! [N: 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))),N)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% sin_cos_npi
tff(fact_9556_sincos__principal__value,axiom,
    ! [X3: real] :
    ? [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,X3) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,X3) ) ) ).

% sincos_principal_value
tff(fact_9557_sin__zero__abs__cos__one,axiom,
    ! [X3: real] :
      ( ( aa_real_real(sin_real,X3) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,X3)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_9558_sin__x__le__x,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X3)),X3)) ) ).

% sin_x_le_x
tff(fact_9559_sin__le__one,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X3)),one_one_real)) ).

% sin_le_one
tff(fact_9560_cos__le__one,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X3)),one_one_real)) ).

% cos_le_one
tff(fact_9561_abs__sin__x__le__abs__x,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X3))),aa_real_real(abs_abs_real,X3))) ).

% abs_sin_x_le_abs_x
tff(fact_9562_cos__arctan__not__zero,axiom,
    ! [X3: real] : aa_real_real(cos_real,aa_real_real(arctan,X3)) != zero_zero_real ).

% cos_arctan_not_zero
tff(fact_9563_sin__cos__le1,axiom,
    ! [X3: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X3)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X3)),aa_real_real(cos_real,Y))))),one_one_real)) ).

% sin_cos_le1
tff(fact_9564_sin__gt__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X3))) ) ) ).

% sin_gt_zero
tff(fact_9565_sin__x__ge__neg__x,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X3)),aa_real_real(sin_real,X3))) ) ).

% sin_x_ge_neg_x
tff(fact_9566_sin__ge__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X3))) ) ) ).

% sin_ge_zero
tff(fact_9567_sin__ge__minus__one,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,X3))) ).

% sin_ge_minus_one
tff(fact_9568_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X3)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_9569_cos__mono__le__eq,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X3)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_9570_cos__inj__pi,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( ( aa_real_real(cos_real,X3) = aa_real_real(cos_real,Y) )
             => ( X3 = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_9571_cos__ge__minus__one,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,X3))) ).

% cos_ge_minus_one
tff(fact_9572_abs__sin__le__one,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X3))),one_one_real)) ).

% abs_sin_le_one
tff(fact_9573_abs__cos__le__one,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,X3))),one_one_real)) ).

% abs_cos_le_one
tff(fact_9574_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_9575_cos__mono__less__eq,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X3)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_9576_cos__monotone__0__pi,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X3)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_9577_sin__eq__0__pi,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),pi))
       => ( ( aa_real_real(sin_real,X3) = zero_zero_real )
         => ( X3 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_9578_sin__zero__pi__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X3)),pi))
     => ( ( aa_real_real(sin_real,X3) = zero_zero_real )
      <=> ( X3 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_9579_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X3))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_9580_sin__zero__iff__int2,axiom,
    ! [X3: real] :
      ( ( aa_real_real(sin_real,X3) = zero_zero_real )
    <=> ? [I3: int] : X3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),pi) ) ).

% sin_zero_iff_int2
tff(fact_9581_sincos__total__pi,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),pi))
            & ( X3 = aa_real_real(cos_real,T6) )
            & ( Y = aa_real_real(sin_real,T6) ) ) ) ) ).

% sincos_total_pi
tff(fact_9582_sin__expansion__lemma,axiom,
    ! [X3: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),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,X3),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_9583_cos__expansion__lemma,axiom,
    ! [X3: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),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,X3),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_9584_sin__gt__zero__02,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X3))) ) ) ).

% sin_gt_zero_02
tff(fact_9585_cos__two__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_less_zero
tff(fact_9586_cos__is__zero,axiom,
    ? [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(cos_real,X) = zero_zero_real )
      & ! [Y4: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X ) ) ) ).

% cos_is_zero
tff(fact_9587_cos__two__le__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_le_zero
tff(fact_9588_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X3))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_9589_cos__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
            & ( aa_real_real(cos_real,X) = Y )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),pi))
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% cos_total
tff(fact_9590_sincos__total__pi__half,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T6: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
              & ( X3 = aa_real_real(cos_real,T6) )
              & ( Y = aa_real_real(sin_real,T6) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_9591_sincos__total__2pi__le,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
          & ( X3 = aa_real_real(cos_real,T6) )
          & ( Y = aa_real_real(sin_real,T6) ) ) ) ).

% sincos_total_2pi_le
tff(fact_9592_sincos__total__2pi,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( ( X3 = aa_real_real(cos_real,T6) )
               => ( Y != aa_real_real(sin_real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_9593_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_9594_sin__gt__zero2,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X3))) ) ) ).

% sin_gt_zero2
tff(fact_9595_sin__lt__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X3)),zero_zero_real)) ) ) ).

% sin_lt_zero
tff(fact_9596_cos__double__less__one,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X3))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_9597_cos__gt__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X3))) ) ) ).

% cos_gt_zero
tff(fact_9598_sin__monotone__2pi__le,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X3))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_9599_sin__mono__le__eq,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X3)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_9600_sin__inj__pi,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( ( aa_real_real(sin_real,X3) = aa_real_real(sin_real,Y) )
             => ( X3 = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_9601_sin__le__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X3)),zero_zero_real)) ) ) ).

% sin_le_zero
tff(fact_9602_sin__less__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X3)),zero_zero_real)) ) ) ).

% sin_less_zero
tff(fact_9603_sin__mono__less__eq,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X3)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_9604_sin__monotone__2pi,axiom,
    ! [Y: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X3))) ) ) ) ).

% sin_monotone_2pi
tff(fact_9605_sin__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9606_cos__gt__zero__pi,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X3))) ) ) ).

% cos_gt_zero_pi
tff(fact_9607_cos__ge__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X3))) ) ) ).

% cos_ge_zero
tff(fact_9608_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_9609_sin__zero__iff__int,axiom,
    ! [X3: real] :
      ( ( aa_real_real(sin_real,X3) = zero_zero_real )
    <=> ? [I3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3))
          & ( X3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_9610_cos__zero__iff__int,axiom,
    ! [X3: real] :
      ( ( aa_real_real(cos_real,X3) = zero_zero_real )
    <=> ? [I3: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3))
          & ( X3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_9611_sin__zero__lemma,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( ( aa_real_real(sin_real,X3) = zero_zero_real )
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & ( X3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_9612_sin__zero__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(sin_real,X3) = zero_zero_real )
    <=> ( ? [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))
            & ( X3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))
            & ( X3 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_9613_cos__zero__lemma,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( ( aa_real_real(cos_real,X3) = zero_zero_real )
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & ( X3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_9614_cos__zero__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(cos_real,X3) = zero_zero_real )
    <=> ( ? [N6: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))
            & ( X3 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N6: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))
            & ( X3 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_9615_Maclaurin__sin__expansion,axiom,
    ! [X3: real,N: nat] :
    ? [T6: real] : aa_real_real(sin_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aji(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X3),N))) ).

% Maclaurin_sin_expansion
tff(fact_9616_Maclaurin__sin__expansion2,axiom,
    ! [X3: real,N: nat] :
    ? [T6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X3)))
      & ( aa_real_real(sin_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aji(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X3),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9617_Maclaurin__cos__expansion,axiom,
    ! [X3: real,N: nat] :
    ? [T6: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,X3)))
      & ( aa_real_real(cos_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ajh(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X3),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9618_Maclaurin__sin__expansion4,axiom,
    ! [X3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ? [T6: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T6))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),X3))
          & ( aa_real_real(sin_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aji(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))),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,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(X3),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_9619_lowi__def,axiom,
    ! [X3: nat,N: nat] : vEBT_VEBT_lowi(X3,N) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% lowi_def
tff(fact_9620_highi__def,axiom,
    ! [X3: nat,N: nat] : vEBT_VEBT_highi(X3,N) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% highi_def
tff(fact_9621_highsimp,axiom,
    ! [X3: nat,N: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N)) = vEBT_VEBT_highi(X3,N) ).

% highsimp
tff(fact_9622_lowsimp,axiom,
    ! [X3: nat,N: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)) = vEBT_VEBT_lowi(X3,N) ).

% lowsimp
tff(fact_9623_summable__arctan__series,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => summable_real(aTP_Lamm_aja(X3)) ) ).

% summable_arctan_series
tff(fact_9624_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),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(N),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_9625_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),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(N),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_9626_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),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(N),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_9627_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),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(N),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_9628_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_adi(F))
     => summable_real(F) ) ).

% summable_rabs_cancel
tff(fact_9629_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N9: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_adi(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_9630_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_adi(F))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_adi(F)))) ) ).

% summable_rabs
tff(fact_9631_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_9632_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_9633_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_9634_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_9635_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real))
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Z))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),one_one_real))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ajj(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_9636_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_9637_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_9638_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_9639_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_9640_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),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))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_9641_tanh__ln__real,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X3)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),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(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_9642_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_9643_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_9644_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% unset_bit_negative_int_iff
tff(fact_9645_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% flip_bit_negative_int_iff
tff(fact_9646_tanh__real__zero__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(tanh_real,X3) = zero_zero_real )
    <=> ( X3 = zero_zero_real ) ) ).

% tanh_real_zero_iff
tff(fact_9647_tanh__real__le__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X3)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ).

% tanh_real_le_iff
tff(fact_9648_tanh__real__pos__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3)) ) ).

% tanh_real_pos_iff
tff(fact_9649_tanh__real__neg__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real)) ) ).

% tanh_real_neg_iff
tff(fact_9650_tanh__real__nonpos__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real)) ) ).

% tanh_real_nonpos_iff
tff(fact_9651_tanh__real__nonneg__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3)) ) ).

% tanh_real_nonneg_iff
tff(fact_9652_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),K)) ).

% unset_bit_less_eq
tff(fact_9653_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = one_one_Code_integer ) ) ) ) ) ).

% sgn_integer_code
tff(fact_9654_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T6: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T6)),aa_real_real(sin_real,T6)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9655_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_9656_tan__npi,axiom,
    ! [N: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_9657_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W2) )
    <=> ( ( A = aa_num_real(numeral_numeral_real,W2) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_9658_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
tff(fact_9659_zero__complex_Ocode,axiom,
    zero_zero_complex = aa_real_complex(complex2(zero_zero_real),zero_zero_real) ).

% zero_complex.code
tff(fact_9660_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_9661_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_9662_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_9663_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9664_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_9665_less__eq__integer__code_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_eq_integer_code(1)
tff(fact_9666_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_9667_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_9668_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_9669_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_9670_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(tan_real,X))) ) ) ).

% lemma_tan_total
tff(fact_9671_tan__gt__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tan_real,X3))) ) ) ).

% tan_gt_zero
tff(fact_9672_tan__pos__pi2__le,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X3))) ) ) ).

% tan_pos_pi2_le
tff(fact_9673_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [X: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(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_9674_tan__less__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X3)),zero_zero_real)) ) ) ).

% tan_less_zero
tff(fact_9675_tan__mono__le,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y))) ) ) ) ).

% tan_mono_le
tff(fact_9676_tan__mono__le__eq,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X3)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_9677_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = K ) ) ) ).

% abs_integer_code
tff(fact_9678_less__integer__code_I1_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_integer_code(1)
tff(fact_9679_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(K),zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9680_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(zero_z3403309356797280102nteger),L) = L ).

% plus_integer_code(2)
tff(fact_9681_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_9682_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(K),zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9683_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( ( ( K = zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = zero_z3403309356797280102nteger ) )
          & ( ( K != zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_9684_sin__paired,axiom,
    ! [X3: real] : pp(aa_real_bool(sums_real(aTP_Lamm_ajk(X3)),aa_real_real(sin_real,X3))) ).

% sin_paired
tff(fact_9685_ceiling__log__eq__powr__iff,axiom,
    ! [X3: real,B: real,K: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(B),X3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),X3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),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_9686_powr__gt__zero,axiom,
    ! [X3: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(powr_real(X3),A)))
    <=> ( X3 != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_9687_powr__nonneg__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(A),X3)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_9688_powr__eq__one__iff,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_real_real(powr_real(A),X3) = one_one_real )
      <=> ( X3 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_9689_powr__one__gt__zero__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(powr_real(X3),one_one_real) = X3 )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3)) ) ).

% powr_one_gt_zero_iff
tff(fact_9690_powr__one,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( aa_real_real(powr_real(X3),one_one_real) = X3 ) ) ).

% powr_one
tff(fact_9691_powr__le__cancel__iff,axiom,
    ! [X3: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X3),A)),aa_real_real(powr_real(X3),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% powr_le_cancel_iff
tff(fact_9692_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9693_powr__log__cancel,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
         => ( aa_real_real(powr_real(A),aa_real_real(log(A),X3)) = X3 ) ) ) ) ).

% powr_log_cancel
tff(fact_9694_powr__numeral,axiom,
    ! [X3: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( aa_real_real(powr_real(X3),aa_num_real(numeral_numeral_real,N)) = aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,N)) ) ) ).

% powr_numeral
tff(fact_9695_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X3: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X3)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X3)) ) ).

% less_eq_integer.abs_eq
tff(fact_9696_powr__mono,axiom,
    ! [A: real,B: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X3),A)),aa_real_real(powr_real(X3),B))) ) ) ).

% powr_mono
tff(fact_9697_powr__mono2,axiom,
    ! [A: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X3),A)),aa_real_real(powr_real(Y),A))) ) ) ) ).

% powr_mono2
tff(fact_9698_powr__ge__pzero,axiom,
    ! [X3: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(powr_real(X3),Y))) ).

% powr_ge_pzero
tff(fact_9699_powr__less__mono2__neg,axiom,
    ! [A: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(Y),A)),aa_real_real(powr_real(X3),A))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_9700_powr__non__neg,axiom,
    ! [A: real,X3: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(A),X3)),zero_zero_real)) ).

% powr_non_neg
tff(fact_9701_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_9702_powr__mono2_H,axiom,
    ! [A: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(Y),A)),aa_real_real(powr_real(X3),A))) ) ) ) ).

% powr_mono2'
tff(fact_9703_powr__less__mono2,axiom,
    ! [A: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(X3),A)),aa_real_real(powr_real(Y),A))) ) ) ) ).

% powr_less_mono2
tff(fact_9704_powr__inj,axiom,
    ! [A: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( ( aa_real_real(powr_real(A),X3) = aa_real_real(powr_real(A),Y) )
        <=> ( X3 = Y ) ) ) ) ).

% powr_inj
tff(fact_9705_gr__one__powr,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(powr_real(X3),Y))) ) ) ).

% gr_one_powr
tff(fact_9706_ge__one__powr__ge__zero,axiom,
    ! [X3: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(powr_real(X3),A))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_9707_powr__mono__both,axiom,
    ! [A: real,B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X3),A)),aa_real_real(powr_real(Y),B))) ) ) ) ) ).

% powr_mono_both
tff(fact_9708_powr__le1,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(X3),A)),one_one_real)) ) ) ) ).

% powr_le1
tff(fact_9709_powr__divide,axiom,
    ! [X3: real,Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(powr_real(aa_real_real(divide_divide_real(X3),Y)),A) = aa_real_real(divide_divide_real(aa_real_real(powr_real(X3),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_divide
tff(fact_9710_powr__mult,axiom,
    ! [X3: real,Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(X3),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_mult
tff(fact_9711_log__base__powr,axiom,
    ! [A: real,B: real,X3: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(log(aa_real_real(powr_real(A),B)),X3) = aa_real_real(divide_divide_real(aa_real_real(log(A),X3)),B) ) ) ).

% log_base_powr
tff(fact_9712_log__powr,axiom,
    ! [X3: real,B: real,Y: real] :
      ( ( X3 != zero_zero_real )
     => ( aa_real_real(log(B),aa_real_real(powr_real(X3),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(log(B),X3)) ) ) ).

% log_powr
tff(fact_9713_ln__powr,axiom,
    ! [X3: real,Y: real] :
      ( ( X3 != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(powr_real(X3),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,X3)) ) ) ).

% ln_powr
tff(fact_9714_powr__realpow,axiom,
    ! [X3: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(powr_real(X3),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(power_power_real(X3),N) ) ) ).

% powr_realpow
tff(fact_9715_less__log__iff,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(log(B),X3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(B),Y)),X3)) ) ) ) ).

% less_log_iff
tff(fact_9716_log__less__iff,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(B),X3)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(powr_real(B),Y))) ) ) ) ).

% log_less_iff
tff(fact_9717_less__powr__iff,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(powr_real(B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(log(B),X3)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_9718_powr__less__iff,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(powr_real(B),Y)),X3))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(log(B),X3))) ) ) ) ).

% powr_less_iff
tff(fact_9719_powr__neg__one,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(powr_real(X3),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(divide_divide_real(one_one_real),X3) ) ) ).

% powr_neg_one
tff(fact_9720_powr__mult__base,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),aa_real_real(powr_real(X3),Y)) = aa_real_real(powr_real(X3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_9721_powr__le__iff,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(B),Y)),X3))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(log(B),X3))) ) ) ) ).

% powr_le_iff
tff(fact_9722_le__powr__iff,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(powr_real(B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(B),X3)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_9723_log__le__iff,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(log(B),X3)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(powr_real(B),Y))) ) ) ) ).

% log_le_iff
tff(fact_9724_le__log__iff,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(log(B),X3)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(B),Y)),X3)) ) ) ) ).

% le_log_iff
tff(fact_9725_ln__powr__bound,axiom,
    ! [X3: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X3)),aa_real_real(divide_divide_real(aa_real_real(powr_real(X3),A)),A))) ) ) ).

% ln_powr_bound
tff(fact_9726_ln__powr__bound2,axiom,
    ! [X3: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(aa_real_real(ln_ln_real,X3)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(A),A)),X3))) ) ) ).

% ln_powr_bound2
tff(fact_9727_add__log__eq__powr,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(log(B),X3)) = aa_real_real(log(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(B),Y)),X3)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_9728_log__add__eq__powr,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(B),X3)),Y) = aa_real_real(log(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),aa_real_real(powr_real(B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_9729_minus__log__eq__powr,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
         => ( aa_real_real(minus_minus_real(Y),aa_real_real(log(B),X3)) = aa_real_real(log(B),aa_real_real(divide_divide_real(aa_real_real(powr_real(B),Y)),X3)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_9730_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_ajl),one_one_real)) ).

% power_half_series
tff(fact_9731_log__minus__eq__powr,axiom,
    ! [B: real,X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
         => ( aa_real_real(minus_minus_real(aa_real_real(log(B),X3)),Y) = aa_real_real(log(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_9732_sums__if_H,axiom,
    ! [G: fun_nat_real,X3: real] :
      ( pp(aa_real_bool(sums_real(G),X3))
     => pp(aa_real_bool(sums_real(aTP_Lamm_ajm(G)),X3)) ) ).

% sums_if'
tff(fact_9733_sums__if,axiom,
    ! [G: fun_nat_real,X3: real,F: fun_nat_real,Y: real] :
      ( pp(aa_real_bool(sums_real(G),X3))
     => ( pp(aa_real_bool(sums_real(F),Y))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ajn(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y))) ) ) ).

% sums_if
tff(fact_9734_powr__neg__numeral,axiom,
    ! [X3: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(powr_real(X3),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,N))) ) ) ).

% powr_neg_numeral
tff(fact_9735_powr__int,axiom,
    ! [X3: real,I: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(powr_real(X3),aa_int_real(ring_1_of_int_real,I)) = aa_nat_real(power_power_real(X3),aa_int_nat(nat2,I)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(powr_real(X3),aa_int_real(ring_1_of_int_real,I)) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(power_power_real(X3),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I)))) ) ) ) ) ).

% powr_int
tff(fact_9736_cos__paired,axiom,
    ! [X3: real] : pp(aa_real_bool(sums_real(aTP_Lamm_ajo(X3)),aa_real_real(cos_real,X3))) ).

% cos_paired
tff(fact_9737_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_9738_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9739_nat__floor__neg,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X3)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9740_floor__eq3,axiom,
    ! [N: nat,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X3)) = N ) ) ) ).

% floor_eq3
tff(fact_9741_le__nat__floor,axiom,
    ! [X3: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X3)),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A)))) ) ).

% le_nat_floor
tff(fact_9742_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_ge
tff(fact_9743_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D2: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D2),E))
         => ( pp(aa_real_bool(P,D2))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_9744_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
    <=> ? [N6: nat] :
          ( ( N6 != zero_zero_nat )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N6))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N6))),E2)) ) ) ).

% real_arch_inverse
tff(fact_9745_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D2: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D2),E))
         => ( pp(aa_real_bool(P,D2))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_9746_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(minus_minus_real(R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_9747_ln__inverse,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X3)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X3)) ) ) ).

% ln_inverse
tff(fact_9748_floor__eq4,axiom,
    ! [N: nat,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X3)) = N ) ) ) ).

% floor_eq4
tff(fact_9749_floor__eq2,axiom,
    ! [N: int,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X3) = N ) ) ) ).

% floor_eq2
tff(fact_9750_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_9751_log__inverse,axiom,
    ! [A: real,X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
         => ( aa_real_real(log(A),aa_real_real(inverse_inverse_real,X3)) = aa_real_real(uminus_uminus_real,aa_real_real(log(A),X3)) ) ) ) ) ).

% log_inverse
tff(fact_9752_exp__plus__inverse__exp,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X3)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X3))))) ).

% exp_plus_inverse_exp
tff(fact_9753_plus__inverse__ge__2,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),aa_real_real(inverse_inverse_real,X3)))) ) ).

% plus_inverse_ge_2
tff(fact_9754_real__le__x__sinh,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X3)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X3)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% real_le_x_sinh
tff(fact_9755_real__le__abs__sinh,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X3)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X3)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% real_le_abs_sinh
tff(fact_9756_floor__log__eq__powr__iff,axiom,
    ! [X3: real,B: real,K: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(B),X3)) = K )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(powr_real(B),aa_int_real(ring_1_of_int_real,K))),X3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),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_9757_powr__real__of__int,axiom,
    ! [X3: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(powr_real(X3),aa_int_real(ring_1_of_int_real,N)) = aa_nat_real(power_power_real(X3),aa_int_nat(nat2,N)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(powr_real(X3),aa_int_real(ring_1_of_int_real,N)) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(X3),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_9758_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9759_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9760_Maclaurin__sin__bound,axiom,
    ! [X3: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(sin_real,X3)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_aji(X3)),aa_nat_set_nat(set_ord_lessThan_nat,N))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,X3)),N)))) ).

% Maclaurin_sin_bound
tff(fact_9761_sinh__real__zero__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(sinh_real,X3) = zero_zero_real )
    <=> ( X3 = zero_zero_real ) ) ).

% sinh_real_zero_iff
tff(fact_9762_sinh__real__le__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X3)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ).

% sinh_real_le_iff
tff(fact_9763_sinh__real__neg__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real)) ) ).

% sinh_real_neg_iff
tff(fact_9764_sinh__real__pos__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3)) ) ).

% sinh_real_pos_iff
tff(fact_9765_sinh__real__nonpos__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real)) ) ).

% sinh_real_nonpos_iff
tff(fact_9766_sinh__real__nonneg__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X3)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3)) ) ).

% sinh_real_nonneg_iff
tff(fact_9767_cosh__real__ge__1,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X3))) ).

% cosh_real_ge_1
tff(fact_9768_cosh__real__nonzero,axiom,
    ! [X3: real] : aa_real_real(cosh_real,X3) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_9769_sinh__le__cosh__real,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X3)),aa_real_real(cosh_real,X3))) ).

% sinh_le_cosh_real
tff(fact_9770_cosh__real__nonneg,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X3))) ).

% cosh_real_nonneg
tff(fact_9771_cosh__real__nonneg__le__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X3)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9772_cosh__real__nonpos__le__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X3)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9773_arcosh__cosh__real,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,X3)) = X3 ) ) ).

% arcosh_cosh_real
tff(fact_9774_cosh__real__pos,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X3))) ).

% cosh_real_pos
tff(fact_9775_cosh__real__strict__mono,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X3)),aa_real_real(cosh_real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_9776_cosh__real__nonneg__less__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X3)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9777_cosh__real__nonpos__less__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X3)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9778_cosh__ln__real,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X3)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),aa_real_real(inverse_inverse_real,X3))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9779_sinh__ln__real,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X3)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X3),aa_real_real(inverse_inverse_real,X3))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9780_cot__less__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X3)),zero_zero_real)) ) ) ).

% cot_less_zero
tff(fact_9781_le__arcsin__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(arcsin,X3)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),X3)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9782_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_9783_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_9784_sin__arcsin,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_9785_cot__npi,axiom,
    ! [N: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_9786_arcsin__le__arcsin,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X3)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_9787_arcsin__minus,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,X3)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,X3)) ) ) ) ).

% arcsin_minus
tff(fact_9788_arcsin__le__mono,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X3)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ) ) ).

% arcsin_le_mono
tff(fact_9789_arcsin__eq__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( ( aa_real_real(arcsin,X3) = aa_real_real(arcsin,Y) )
        <=> ( X3 = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_9790_arcsin__less__arcsin,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X3)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9791_arcsin__less__mono,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X3)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_9792_cos__arcsin__nonzero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X3)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9793_arcsin__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))) ) ) ).

% arcsin_lbound
tff(fact_9794_arcsin__ubound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arcsin_ubound
tff(fact_9795_arcsin__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_bounded
tff(fact_9796_cot__gt__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cot_real,X3))) ) ) ).

% cot_gt_zero
tff(fact_9797_arcsin__sin,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),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,X3)) = X3 ) ) ) ).

% arcsin_sin
tff(fact_9798_arcsin,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(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_9799_arcsin__pi,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),pi))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_9800_arcsin__le__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X3)),Y))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(sin_real,Y))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9801_i__even__power,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).

% i_even_power
tff(fact_9802_set__decode__0,axiom,
    ! [X3: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,X3)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3)) ) ).

% set_decode_0
tff(fact_9803_set__decode__inverse,axiom,
    ! [N: nat] : aa_set_nat_nat(nat_set_encode,aa_nat_set_nat(nat_set_decode,N)) = N ).

% set_decode_inverse
tff(fact_9804_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).

% Suc_0_mod_eq
tff(fact_9805_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_9806_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_9807_set__decode__Suc,axiom,
    ! [N: nat,X3: nat] :
      ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(nat_set_decode,X3)))
    <=> pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% set_decode_Suc
tff(fact_9808_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
tff(fact_9809_finite__set__decode,axiom,
    ! [N: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(nat_set_decode,N))) ).

% finite_set_decode
tff(fact_9810_Complex__eq__i,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_complex(complex2(X3),Y) = imaginary_unit )
    <=> ( ( X3 = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9811_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9812_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% subset_decode_imp_le
tff(fact_9813_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4048646594741130792ol_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_list_bool_nat(size_size_list_bool,Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_9814_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_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9815_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,Z)))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Z)) = aa_set_nat_set_nat(insert_nat(N),aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_9816_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_9817_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_9818_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_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_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) )
          & ( ( aa_int_int(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(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_9819_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) )
      & ( ( 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(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_9820_set__decode__def,axiom,
    ! [X3: nat] : aa_nat_set_nat(nat_set_decode,X3) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ajp,X3)) ).

% set_decode_def
tff(fact_9821_and__int_Opelims,axiom,
    ! [X3: int,Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X3),Xa) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Xa)))
       => ~ ( ( ( ( pp(aa_set_int_bool(member_int(X3),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))))
                  & pp(aa_set_int_bool(member_int(Xa),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))) )
               => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X3))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))) ) )
              & ( ~ ( pp(aa_set_int_bool(member_int(X3),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(Xa),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))) )
               => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X3))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Xa))) ) ) ) ).

% and_int.pelims
tff(fact_9822_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K),L)))
     => ( ( ( pp(aa_set_int_bool(member_int(K),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(K),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(L),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_9823_and__int_Oelims,axiom,
    ! [X3: int,Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X3),Xa) = Y )
     => ( ( ( pp(aa_set_int_bool(member_int(X3),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(Xa),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))) )
         => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X3))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(X3),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(Xa),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))) )
         => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X3))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_9824_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_9825_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% and_negative_int_iff
tff(fact_9826_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_9827_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_9828_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X3),Y)),Z)) ) ) ).

% AND_upper2'
tff(fact_9829_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_9830_AND__upper2,axiom,
    ! [Y: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X3),Y)),Y)) ) ).

% AND_upper2
tff(fact_9831_AND__upper1,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X3),Y)),X3)) ) ).

% AND_upper1
tff(fact_9832_AND__lower,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X3),Y))) ) ).

% AND_lower
tff(fact_9833_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X3),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_9834_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_9835_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K)) ) ).

% and_less_eq
tff(fact_9836_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero_int )
          | ( L = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = zero_zero_int ) )
      & ( ~ ( ( K = zero_zero_int )
            | ( L = zero_zero_int ) )
       => ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = L ) )
          & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = K ) )
              & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_9837_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa_set_int_bool(member_int(K),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))))
          & pp(aa_set_int_bool(member_int(L),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
      & ( ~ ( pp(aa_set_int_bool(member_int(K),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),aa_set_int_set_int(insert_int(zero_zero_int),aa_set_int_set_int(insert_int(aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_9838_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] : adjust_div(aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q4),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)))) ).

% Divides.adjust_div_eq
tff(fact_9839_uint32_Osize__eq,axiom,
    ! [P3: uint32] : size_size_uint32(P3) = 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)))))) ).

% uint32.size_eq
tff(fact_9840_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_9841_and__nat__numerals_I3_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_9842_and__nat__numerals_I4_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_9843_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_9844_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9845_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9846_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( ( M = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_9847_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_9848_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_ajq(L),Qr) ).

% divmod_step_nat_def
tff(fact_9849_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_ajr(L),Qr) ).

% divmod_step_int_def
tff(fact_9850_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ajs(L),Qr) ).

% divmod_step_integer_def
tff(fact_9851_False__map2__and,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs)),aa_set_bool_set_bool(insert_bool(fFalse),bot_bot_set_bool)))
     => ( ( aa_list_bool_nat(size_size_list_bool,Ys) = aa_list_bool_nat(size_size_list_bool,Xs) )
       => ( map_Pr8574941368599290932l_bool(produc8645476572806212523l_bool(fconj),zip_bool_bool(Xs,Ys)) = Xs ) ) ) ).

% False_map2_and
tff(fact_9852_False__map2__or,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Xs)),aa_set_bool_set_bool(insert_bool(fFalse),bot_bot_set_bool)))
     => ( ( aa_list_bool_nat(size_size_list_bool,Ys) = aa_list_bool_nat(size_size_list_bool,Xs) )
       => ( map_Pr8574941368599290932l_bool(produc8645476572806212523l_bool(fdisj),zip_bool_bool(Xs,Ys)) = Ys ) ) ) ).

% False_map2_or
tff(fact_9853_align__lem__or,axiom,
    ! [Xs: list_bool,N: nat,M: nat,Ys: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
     => ( ( aa_list_bool_nat(size_size_list_bool,Ys) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
       => ( ( drop_bool(M,Xs) = replicate_bool(N,fFalse) )
         => ( ( take_bool(M,Ys) = replicate_bool(M,fFalse) )
           => ( map_Pr8574941368599290932l_bool(produc8645476572806212523l_bool(fdisj),zip_bool_bool(Xs,Ys)) = append_bool(take_bool(M,Xs),drop_bool(M,Ys)) ) ) ) ) ) ).

% align_lem_or
tff(fact_9854_align__lem__and,axiom,
    ! [Xs: list_bool,N: nat,M: nat,Ys: list_bool] :
      ( ( aa_list_bool_nat(size_size_list_bool,Xs) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
     => ( ( aa_list_bool_nat(size_size_list_bool,Ys) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
       => ( ( drop_bool(M,Xs) = replicate_bool(N,fFalse) )
         => ( ( take_bool(M,Ys) = replicate_bool(M,fFalse) )
           => ( map_Pr8574941368599290932l_bool(produc8645476572806212523l_bool(fconj),zip_bool_bool(Xs,Ys)) = replicate_bool(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),fFalse) ) ) ) ) ) ).

% align_lem_and
tff(fact_9855_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_ajt) ).

% prod_encode_def
tff(fact_9856_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_aju,Qr) ).

% Divides.adjust_div_def
tff(fact_9857_sort__upt,axiom,
    ! [M: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_dl,upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_9858_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_dm,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_9859_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = produc2626176000494625587at_nat(aTP_Lamm_ajv,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_9860_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_ajw(D))) ).

% int_ge_less_than_def
tff(fact_9861_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_ajx(D))) ).

% int_ge_less_than2_def
tff(fact_9862_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% mask_nat_positive_iff
tff(fact_9863_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),zero_zero_int)) ).

% not_mask_negative_int
tff(fact_9864_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).

% mask_nonnegative_int
tff(fact_9865_less__eq__mask,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% less_eq_mask
tff(fact_9866_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).

% less_mask
tff(fact_9867_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_mask_eq_exp
tff(fact_9868_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% mask_nat_less_exp
tff(fact_9869_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ajy,Y)) ).

% arctan_def
tff(fact_9870_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ajz,Y)) ).

% arcsin_def
tff(fact_9871_ln__real__def,axiom,
    ! [X3: real] : aa_real_real(ln_ln_real,X3) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_aka,X3)) ).

% ln_real_def
tff(fact_9872_ln__neg__is__const,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real))
     => ( aa_real_real(ln_ln_real,X3) = the_real(aTP_Lamm_ba) ) ) ).

% ln_neg_is_const
tff(fact_9873_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_akb) ).

% pi_half
tff(fact_9874_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_akb)) ).

% pi_def
tff(fact_9875_floor__real__def,axiom,
    ! [X3: real] : aa_real_int(archim6058952711729229775r_real,X3) = the_int(aa_real_fun_int_bool(aTP_Lamm_akc,X3)) ).

% floor_real_def
tff(fact_9876_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_9877_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_9878_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_9879_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9880_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(N),K))) ).

% take_bit_nonnegative
tff(fact_9881_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_9882_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q4))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9883_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_9884_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9885_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9886_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_9887_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9888_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9889_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% take_bit_nat_less_exp
tff(fact_9890_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_9891_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_9892_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),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_9893_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9894_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_9895_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9896_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_9897_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_9898_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_9899_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_9900_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9901_floor__rat__def,axiom,
    ! [X3: rat] : aa_rat_int(archim3151403230148437115or_rat,X3) = the_int(aTP_Lamm_akd(X3)) ).

% floor_rat_def
tff(fact_9902_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),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_9903_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% not_nonnegative_int_iff
tff(fact_9904_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% not_negative_int_iff
tff(fact_9905_less__eq__rat__def,axiom,
    ! [X3: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Y))
        | ( X3 = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9906_abs__rat__def,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_rat_def
tff(fact_9907_sgn__rat__def,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_9908_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
     => ~ ! [S: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),S))
           => ! [T6: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T6))
               => ( R2 != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,S),T6) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9909_and__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9910_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9911_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_9912_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
    <=> ( aa_num_nat(pred_numeral,K) = N ) ) ).

% eq_numeral_Suc
tff(fact_9913_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( N = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_9914_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% less_numeral_Suc
tff(fact_9915_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% less_Suc_numeral
tff(fact_9916_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% le_Suc_numeral
tff(fact_9917_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% le_numeral_Suc
tff(fact_9918_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_9919_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).

% diff_numeral_Suc
tff(fact_9920_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_9921_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N)) ).

% max_numeral_Suc
tff(fact_9922_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_9923_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_num_nat(numeral_numeral_nat,K)) = aa_set_nat_set_nat(insert_nat(aa_num_nat(pred_numeral,K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_num_nat(pred_numeral,K))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_9924_int__not__code_I1_J,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_not_code(1)
tff(fact_9925_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,L) ) )
      & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,K) ) )
          & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( K = zero_zero_int )
               => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = L ) )
              & ( ( K != zero_zero_int )
               => ( ( ( L = zero_zero_int )
                   => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = K ) )
                  & ( ( L != zero_zero_int )
                   => ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_9926_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9927_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% xor_negative_int_iff
tff(fact_9928_XOR__lower,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X3),Y))) ) ) ).

% XOR_lower
tff(fact_9929_int__xor__code_I1_J,axiom,
    ! [J: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,zero_zero_int),J) = J ).

% int_xor_code(1)
tff(fact_9930_int__xor__code_I2_J,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,I),zero_zero_int) = I ).

% int_xor_code(2)
tff(fact_9931_XOR__upper,axiom,
    ! [X3: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X3),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_9932_int__and__code_I2_J,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,I),zero_zero_int) = zero_zero_int ).

% int_and_code(2)
tff(fact_9933_int__and__code_I1_J,axiom,
    ! [J: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),J) = zero_zero_int ).

% int_and_code(1)
tff(fact_9934_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),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,N))),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9935_arcosh__real__def,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3))
     => ( aa_real_real(arcosh_real,X3) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_9936_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X3: real] :
      ( ( aa_real_real(sqrt,X3) = zero_zero_real )
    <=> ( X3 = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_9937_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_9938_real__sqrt__le__iff,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X3)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y)) ) ).

% real_sqrt_le_iff
tff(fact_9939_real__sqrt__lt__0__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_9940_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ).

% real_sqrt_gt_0_iff
tff(fact_9941_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ).

% real_sqrt_ge_0_iff
tff(fact_9942_real__sqrt__le__0__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X3)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real)) ) ).

% real_sqrt_le_0_iff
tff(fact_9943_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ).

% real_sqrt_ge_1_iff
tff(fact_9944_real__sqrt__le__1__iff,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X3)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real)) ) ).

% real_sqrt_le_1_iff
tff(fact_9945_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9946_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),zero_zero_int))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_9947_xor__nat__numerals_I4_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3)) ).

% xor_nat_numerals(4)
tff(fact_9948_xor__nat__numerals_I3_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3)) ).

% xor_nat_numerals(3)
tff(fact_9949_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_9950_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_9951_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2)))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9952_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W2)))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9953_real__sqrt__pow2__iff,axiom,
    ! [X3: real] :
      ( ( aa_nat_real(power_power_real(aa_real_real(sqrt,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X3 )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3)) ) ).

% real_sqrt_pow2_iff
tff(fact_9954_real__sqrt__pow2,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( aa_nat_real(power_power_real(aa_real_real(sqrt,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X3 ) ) ).

% real_sqrt_pow2
tff(fact_9955_bin__nth__minus__Bit0,axiom,
    ! [N: nat,W2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))),N))
      <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% bin_nth_minus_Bit0
tff(fact_9956_bin__nth__minus__Bit1,axiom,
    ! [N: nat,W2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W2))),N))
      <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% bin_nth_minus_Bit1
tff(fact_9957_test__bit__int__code_I1_J,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,zero_zero_int),N)) ).

% test_bit_int_code(1)
tff(fact_9958_real__sqrt__le__mono,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X3)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_le_mono
tff(fact_9959_real__sqrt__gt__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,X3))) ) ).

% real_sqrt_gt_zero
tff(fact_9960_real__sqrt__ge__one,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,X3))) ) ).

% real_sqrt_ge_one
tff(fact_9961_real__sqrt__ge__zero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,X3))) ) ).

% real_sqrt_ge_zero
tff(fact_9962_real__sqrt__eq__zero__cancel,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( ( aa_real_real(sqrt,X3) = zero_zero_real )
       => ( X3 = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_9963_real__div__sqrt,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( aa_real_real(divide_divide_real(X3),aa_real_real(sqrt,X3)) = aa_real_real(sqrt,X3) ) ) ).

% real_div_sqrt
tff(fact_9964_sqrt__add__le__add__sqrt,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,X3)),aa_real_real(sqrt,Y)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_9965_le__real__sqrt__sumsq,axiom,
    ! [X3: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))) ).

% le_real_sqrt_sumsq
tff(fact_9966_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_int_int(bit_se2923211474154528505it_int(N),L) )
     => ( map_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),upt(zero_zero_nat,N)) = map_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),upt(zero_zero_nat,N)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_9967_sqrt__divide__self__eq,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( aa_real_real(divide_divide_real(aa_real_real(sqrt,X3)),X3) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X3)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9968_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9969_sqrt__le__D,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X3)),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% sqrt_le_D
tff(fact_9970_real__le__rsqrt,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(sqrt,Y))) ) ).

% real_le_rsqrt
tff(fact_9971_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M3))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),M3))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_9972_real__sqrt__unique,axiom,
    ! [Y: real,X3: real] :
      ( ( aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X3 )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(sqrt,X3) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_9973_real__le__lsqrt,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X3)),Y)) ) ) ) ).

% real_le_lsqrt
tff(fact_9974_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),U))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(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_9975_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( X3 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_9976_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X3: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = X3 )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_9977_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(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(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9978_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_9979_real__sqrt__sum__squares__ge1,axiom,
    ! [X3: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_9980_sqrt__ge__absD,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),aa_real_real(sqrt,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)) ) ).

% sqrt_ge_absD
tff(fact_9981_real__less__lsqrt,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X3)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_9982_sqrt__sum__squares__le__sum,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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,X3),Y))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_9983_real__inv__sqrt__pow2,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X3) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9984_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X3: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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,X3)),aa_real_real(abs_abs_real,Y)))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_9985_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs2
tff(fact_9986_real__sqrt__ge__abs1,axiom,
    ! [X3: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_ge_abs1
tff(fact_9987_ln__sqrt,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X3)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_9988_real__sqrt__power__even,axiom,
    ! [N: nat,X3: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
       => ( aa_nat_real(power_power_real(aa_real_real(sqrt,X3)),N) = aa_nat_real(power_power_real(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_9989_arsinh__real__aux,axiom,
    ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_9990_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X3: real,Y: real,Xa: real,Ya: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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(power_power_real(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9991_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_9992_arith__geo__mean__sqrt,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),Y))),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_9993_powr__half__sqrt,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( aa_real_real(powr_real(X3),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,X3) ) ) ).

% powr_half_sqrt
tff(fact_9994_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9995_cos__x__y__le__one,axiom,
    ! [X3: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(X3),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_9996_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% xor_Suc_0_eq
tff(fact_9997_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% Suc_0_xor_eq
tff(fact_9998_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),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))),N)),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% take_bit_Suc_from_most
tff(fact_9999_sqrt__sum__squares__half__less,axiom,
    ! [X3: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_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_10000_sin__cos__sqrt,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X3)))
     => ( aa_real_real(sin_real,X3) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(aa_real_real(cos_real,X3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_10001_cos__arcsin,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X3)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_10002_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_10003_sin__arccos,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X3)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(power_power_real(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_10004_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_10005_cos__arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_10006_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_10007_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_10008_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_10009_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N))) ).

% not_bit_Suc_0_numeral
tff(fact_10010_arccos__le__arccos,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X3))) ) ) ) ).

% arccos_le_arccos
tff(fact_10011_arccos__le__mono,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,X3)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X3)) ) ) ) ).

% arccos_le_mono
tff(fact_10012_arccos__eq__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)) )
     => ( ( aa_real_real(arccos,X3) = aa_real_real(arccos,Y) )
      <=> ( X3 = Y ) ) ) ).

% arccos_eq_iff
tff(fact_10013_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ) ).

% bit_nat_iff
tff(fact_10014_arccos__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))) ) ) ).

% arccos_lbound
tff(fact_10015_arccos__less__arccos,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X3))) ) ) ) ).

% arccos_less_arccos
tff(fact_10016_arccos__less__mono,axiom,
    ! [X3: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X3)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X3)) ) ) ) ).

% arccos_less_mono
tff(fact_10017_arccos__ubound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ).

% arccos_ubound
tff(fact_10018_arccos__cos,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),pi))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X3)) = X3 ) ) ) ).

% arccos_cos
tff(fact_10019_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_10020_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi))
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_10021_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_10022_arccos__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_10023_sin__arccos__nonzero,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X3)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_10024_arccos__cos2,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),X3))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X3)) = aa_real_real(uminus_uminus_real,X3) ) ) ) ).

% arccos_cos2
tff(fact_10025_arccos__minus,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),one_one_real))
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X3)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,X3)) ) ) ) ).

% arccos_minus
tff(fact_10026_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ake,Y)) ).

% arccos_def
tff(fact_10027_arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi))
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_10028_arccos__minus__abs,axiom,
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X3)),one_one_real))
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X3)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,X3)) ) ) ).

% arccos_minus_abs
tff(fact_10029_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arccos_le_pi2
tff(fact_10030_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_10031_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_10032_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_10033_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
          | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) )
      & ( ~ ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
            | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( ( ( K = zero_zero_int )
           => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = L ) )
          & ( ( K != zero_zero_int )
           => ( ( ( L = zero_zero_int )
               => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = K ) )
              & ( ( L != zero_zero_int )
               => ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_10034_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_10035_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% or_negative_int_iff
tff(fact_10036_OR__lower,axiom,
    ! [X3: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X3),Y))) ) ) ).

% OR_lower
tff(fact_10037_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))) ) ).

% or_greater_eq
tff(fact_10038_int__or__code_I2_J,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,I),zero_zero_int) = I ).

% int_or_code(2)
tff(fact_10039_int__or__code_I1_J,axiom,
    ! [J: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,zero_zero_int),J) = J ).

% int_or_code(1)
tff(fact_10040_or__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_10041_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_10042_OR__upper,axiom,
    ! [X3: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X3),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_10043_or__nat__numerals_I4_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3)) ).

% or_nat_numerals(4)
tff(fact_10044_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_10045_or__nat__numerals_I3_J,axiom,
    ! [X3: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X3))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X3)) ).

% or_nat_numerals(3)
tff(fact_10046_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_10047_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_10048_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_10049_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_10050_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_10051_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).

% take_bit_num_simps(1)
tff(fact_10052_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa_nat_nat(suc,N),one) = aa_num_option_num(some_num,one) ).

% take_bit_num_simps(2)
tff(fact_10053_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_akf,bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_10054_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit1,M)) = aa_num_option_num(some_num,case_option_num_num(one,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_10055_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_akf,bit_take_bit_num(aa_num_nat(pred_numeral,R2),M)) ).

% take_bit_num_simps(6)
tff(fact_10056_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,M)) = aa_num_option_num(some_num,case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),M))) ).

% take_bit_num_simps(7)
tff(fact_10057_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,aTP_Lamm_akg(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_10058_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).

% and_minus_numerals(4)
tff(fact_10059_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).

% and_minus_numerals(8)
tff(fact_10060_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_akh,bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_10061_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_10062_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_10063_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_10064_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N))) ).

% and_minus_numerals(3)
tff(fact_10065_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N))) ).

% and_minus_numerals(7)
tff(fact_10066_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_10067_semiring__norm_I27_J,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(bitM,N)) ).

% semiring_norm(27)
tff(fact_10068_semiring__norm_I28_J,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,N)) ).

% semiring_norm(28)
tff(fact_10069_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,N))) ).

% eval_nat_numeral(2)
tff(fact_10070_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit1,M)) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_aki(M)),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_10071_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit0,M)) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_akj(M)),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_10072_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : bit_take_bit_num(N,one) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_akk),N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_10073_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = produc478579273971653890on_num(aTP_Lamm_ako,aa_num5026351239867140595at_num(product_Pair_nat_num(N),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_10074_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_akp,quotient_of(P3)) ).

% rat_inverse_code
tff(fact_10075_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_10076_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_akq),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_10077_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_bc),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_10078_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_10079_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q4: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4)) ) ).

% quotient_of_denom_pos
tff(fact_10080_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_akr,N)),M) ).

% max_Suc1
tff(fact_10081_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_aks,N)),M) ).

% max_Suc2
tff(fact_10082_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_dl),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_10083_rat__uminus__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_akt,quotient_of(P3)) ).

% rat_uminus_code
tff(fact_10084_rat__less__code,axiom,
    ! [P3: rat,Q4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),Q4))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_akv(Q4)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_10085_rat__floor__code,axiom,
    ! [P3: rat] : aa_rat_int(archim3151403230148437115or_rat,P3) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P3)) ).

% rat_floor_code
tff(fact_10086_rat__abs__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_akw,quotient_of(P3)) ).

% rat_abs_code
tff(fact_10087_rat__less__eq__code,axiom,
    ! [P3: rat,Q4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P3),Q4))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_aky(Q4)),quotient_of(P3))) ) ).

% rat_less_eq_code
tff(fact_10088_rat__plus__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P3),Q4)) = produc4245557441103728435nt_int(aTP_Lamm_ala(Q4),quotient_of(P3)) ).

% rat_plus_code
tff(fact_10089_rat__minus__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,P3),Q4)) = produc4245557441103728435nt_int(aTP_Lamm_alc(Q4),quotient_of(P3)) ).

% rat_minus_code
tff(fact_10090_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),zero_zero_int)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_10091_normalize__negative,axiom,
    ! [Q4: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q4),zero_zero_int))
     => ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3)),aa_int_int(uminus_uminus_int,Q4))) ) ) ).

% normalize_negative
tff(fact_10092_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q4: int] :
      ( ( normalize(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4)) ) ).

% normalize_denom_pos
tff(fact_10093_normalize__crossproduct,axiom,
    ! [Q4: int,S2: int,P3: int,R2: int] :
      ( ( Q4 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P3),Q4)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(R2),S2)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P3),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q4) ) ) ) ) ).

% normalize_crossproduct
tff(fact_10094_rat__times__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P3),Q4)) = produc4245557441103728435nt_int(aTP_Lamm_ale(Q4),quotient_of(P3)) ).

% rat_times_code
tff(fact_10095_rat__divide__code,axiom,
    ! [P3: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P3),Q4)) = produc4245557441103728435nt_int(aTP_Lamm_alg(Q4),quotient_of(P3)) ).

% rat_divide_code
tff(fact_10096_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
       => ( bit_take_bit_num(N,M) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
       => ( bit_take_bit_num(N,M) = aa_num_option_num(some_num,aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)))) ) ) ) ).

% take_bit_num_def
tff(fact_10097_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_10098_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% drop_bit_negative_int_iff
tff(fact_10099_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_10100_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_10101_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_10102_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se8568078237143864401it_int(zero_zero_nat),I) = I ).

% drop_bit_int_code(1)
tff(fact_10103_drop__bit__int__code_I2_J,axiom,
    ! [N: nat] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),zero_zero_int) = zero_zero_int ).

% drop_bit_int_code(2)
tff(fact_10104_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_10105_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_10106_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
     => ( aa_nat_num(num_of_nat,N) = one ) ) ).

% num_of_nat_One
tff(fact_10107_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_10108_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_10109_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = aa_num_num(inc,aa_nat_num(num_of_nat,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_10110_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_10111_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se545348938243370406it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% push_bit_negative_int_iff
tff(fact_10112_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% push_bit_of_Suc_0
tff(fact_10113_push__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se545348938243370406it_int(zero_zero_nat),I) = I ).

% push_bit_int_code(1)
tff(fact_10114_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se545348938243370406it_int(M),K)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_10115_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q4)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_10116_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% push_bit_nat_def
tff(fact_10117_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,N)),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(aa_int_fun_int_int(bit_concat_bit(N),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_10118_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K),L) = L ).

% concat_bit_0
tff(fact_10119_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(N),K) ).

% concat_bit_of_zero_2
tff(fact_10120_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_10121_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ).

% concat_bit_negative_iff
tff(fact_10122_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(N),L) ).

% concat_bit_of_zero_1
tff(fact_10123_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),K),L)),N))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_10124_int__sdiv__simps_I2_J,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),zero_zero_int) = zero_zero_int ).

% int_sdiv_simps(2)
tff(fact_10125_sdiv__int__0__div,axiom,
    ! [X3: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,zero_zero_int),X3) = zero_zero_int ).

% sdiv_int_0_div
tff(fact_10126_sdiv__int__div__0,axiom,
    ! [X3: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,X3),zero_zero_int) = zero_zero_int ).

% sdiv_int_div_0
tff(fact_10127_int__sdiv__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B) = A )
      <=> ( B = one_one_int ) ) ) ).

% int_sdiv_same_is_1
tff(fact_10128_int__sdiv__negated__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B) = aa_int_int(uminus_uminus_int,A) )
      <=> ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_10129_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A),B)) = aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_10130_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M5))
         => ! [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N6))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(X6,M5)),aa_nat_real(X6,N6)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_10131_len__num0,axiom,
    ! [Uu: itself_Numeral_num0] : type_l4264026598287037464l_num0(Uu) = zero_zero_nat ).

% len_num0
tff(fact_10132_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).

% min_Suc_Suc
tff(fact_10133_min__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

% min_0L
tff(fact_10134_min__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_10135_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_10136_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_10137_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_10138_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).

% min_numeral_Suc
tff(fact_10139_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)) ).

% nat_mult_min_left
tff(fact_10140_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_min_right
tff(fact_10141_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_10142_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

% min_diff
tff(fact_10143_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_alh,N)),M) ).

% min_Suc1
tff(fact_10144_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_ali,N)),M) ).

% min_Suc2
tff(fact_10145_min__enat__simps_I2_J,axiom,
    ! [Q4: extended_enat] : aa_Ext7969882492699152700d_enat(ord_mi8085742599997312461d_enat(Q4),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_10146_min__enat__simps_I3_J,axiom,
    ! [Q4: extended_enat] : aa_Ext7969882492699152700d_enat(ord_mi8085742599997312461d_enat(zero_z5237406670263579293d_enat),Q4) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_10147_int__set__bits__K__False,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_bc) = zero_zero_int ).

% int_set_bits_K_False
tff(fact_10148_int__set__bits__K__True,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_akq) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_set_bits_K_True
tff(fact_10149_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_10150_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_10151_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_10152_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = aa_set_real_set_real(insert_real(zero_zero_real),bot_bot_set_real) ) )
      & ( ( A != zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = top_top_set_real ) ) ) ).

% range_mult
tff(fact_10153_norm__assertion__simps_I4_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),top_top_assn) = X3 ).

% norm_assertion_simps(4)
tff(fact_10154_norm__assertion__simps_I3_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,top_top_assn),X3) = X3 ).

% norm_assertion_simps(3)
tff(fact_10155_ent__star__mono__true,axiom,
    ! [A2: assn,A8: assn,B3: assn,B10: assn] :
      ( entails(A2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A8),top_top_assn))
     => ( entails(B3,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B10),top_top_assn))
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B3)),top_top_assn),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A8),B10)),top_top_assn)) ) ) ).

% ent_star_mono_true
tff(fact_10156_ent__refl__true,axiom,
    ! [A2: assn] : entails(A2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),top_top_assn)) ).

% ent_refl_true
tff(fact_10157_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)) ) ).

% ent_true_drop(1)
tff(fact_10158_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
     => entails(P,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)) ) ).

% ent_true_drop(2)
tff(fact_10159_merge__true__star__ctx,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),P)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),P) ).

% merge_true_star_ctx
tff(fact_10160_ent__true,axiom,
    ! [P: assn] : entails(P,top_top_assn) ).

% ent_true
tff(fact_10161_norm__assertion__simps_I12_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),top_top_assn) = top_top_assn ).

% norm_assertion_simps(12)
tff(fact_10162_norm__assertion__simps_I11_J,axiom,
    ! [X3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,top_top_assn),X3) = top_top_assn ).

% norm_assertion_simps(11)
tff(fact_10163_UNIV__bool,axiom,
    top_top_set_bool = aa_set_bool_set_bool(insert_bool(fFalse),aa_set_bool_set_bool(insert_bool(fTrue),bot_bot_set_bool)) ).

% UNIV_bool
tff(fact_10164_surj__list__encode,axiom,
    image_list_nat_nat(nat_list_encode,top_top_set_list_nat) = top_top_set_nat ).

% surj_list_encode
tff(fact_10165_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_10166_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(set_ord_lessThan_nat),top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_10167_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(set_ord_atMost_nat),top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_10168_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_10169_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_alj,N)),top_top_set_nat) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_10170_suminf__eq__SUP__real,axiom,
    ! [X6: fun_nat_real] :
      ( summable_real(X6)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(X6,I2)))
       => ( suminf_real(X6) = comple1385675409528146559p_real(aa_set_nat_set_real(image_nat_real(aTP_Lamm_alk(X6)),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_10171_bin__last__set__bits,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(F)
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_bi6516823479961619367ts_int(F)))
      <=> pp(aa_nat_bool(F,zero_zero_nat)) ) ) ).

% bin_last_set_bits
tff(fact_10172_wf__set__bits__int__Suc,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(aTP_Lamm_all(F))
    <=> bit_wf_set_bits_int(F) ) ).

% wf_set_bits_int_Suc
tff(fact_10173_ones,axiom,
    ! [N: nat,F: fun_nat_bool] :
      ( ! [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
         => pp(aa_nat_bool(F,N5)) )
     => bit_wf_set_bits_int(F) ) ).

% ones
tff(fact_10174_wf__set__bits__int_Ocases,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(F)
     => ( ! [N2: nat] :
            ~ ! [N4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N4))
               => ~ pp(aa_nat_bool(F,N4)) )
       => ~ ! [N2: nat] :
              ~ ! [N4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N4))
                 => pp(aa_nat_bool(F,N4)) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_10175_wf__set__bits__int_Osimps,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(F)
    <=> ( ? [N6: nat] :
          ! [N10: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N10))
           => ~ pp(aa_nat_bool(F,N10)) )
        | ? [N6: nat] :
          ! [N10: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N10))
           => pp(aa_nat_bool(F,N10)) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_10176_zeros,axiom,
    ! [N: nat,F: fun_nat_bool] :
      ( ! [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N5))
         => ~ pp(aa_nat_bool(F,N5)) )
     => bit_wf_set_bits_int(F) ) ).

% zeros
tff(fact_10177_wf__set__bits__int__simps,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(F)
    <=> ? [N6: nat] :
          ( ! [N10: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N10))
             => ~ pp(aa_nat_bool(F,N10)) )
          | ! [N10: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N10))
             => pp(aa_nat_bool(F,N10)) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_10178_wf__set__bits__int__const,axiom,
    ! [B: bool] : bit_wf_set_bits_int(aa_bool_fun_nat_bool(aTP_Lamm_io,B)) ).

% wf_set_bits_int_const
tff(fact_10179_int__set__bits__unfold__BIT,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(F)
     => ( bit_bi6516823479961619367ts_int(F) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(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_bool_nat(F,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_10180_bin__rest__set__bits,axiom,
    ! [F: fun_nat_bool] :
      ( 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_bool_nat(F,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_10181_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_10182_shiftl__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh3965577149348748681tl_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% shiftl_Suc_0
tff(fact_10183_shiftr__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh2154871086232339855tr_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% shiftr_Suc_0
tff(fact_10184_card__Collect__less__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cg,N))) = N ).

% card_Collect_less_nat
tff(fact_10185_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_10186_card__Collect__le__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cf,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_10187_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(aa_nat1818925681583257441et_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_10188_binomial__def,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_alm(N),K))) ).

% binomial_def
tff(fact_10189_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(aa_int_set_int(set_or4662586982721622107an_int(zero_zero_int),U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_10190_card__less__Suc2,axiom,
    ! [M8: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M8))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aln(M8),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_alo(M8),I))) ) ) ).

% card_less_Suc2
tff(fact_10191_card__less__Suc,axiom,
    ! [M8: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M8))
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aln(M8),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_alo(M8),I))) ) ) ).

% card_less_Suc
tff(fact_10192_card__less,axiom,
    ! [M8: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M8))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_alo(M8),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_10193_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,A2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2)))))
     => ( A2 = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_10194_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,N7),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N7)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_10195_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S3)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S3)))) ) ).

% card_le_Suc_Max
tff(fact_10196_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dl),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dl),S3))) ).

% card_sum_le_nat_sum
tff(fact_10197_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_alp(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_10198_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_acs,N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_10199_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_alq(M),N7))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_alr(M),N7)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_als(M),N7)))) ) ) ).

% card_length_sum_list_rec
tff(fact_10200_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_alq(M),N7))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),M)),one_one_nat)),N7) ).

% card_length_sum_list
tff(fact_10201_card__num0,axiom,
    finite6454714172617411596l_num0(top_to3689904424835650196l_num0) = zero_zero_nat ).

% card_num0
tff(fact_10202_card__nat,axiom,
    aa_set_nat_nat(finite_card_nat,top_top_set_nat) = zero_zero_nat ).

% card_nat
tff(fact_10203_card__literal,axiom,
    finite_card_literal(top_top_set_literal) = zero_zero_nat ).

% card_literal
tff(fact_10204_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K5)),K5)) ) ).

% Inf_nat_def1
tff(fact_10205_inj__on__set__encode,axiom,
    pp(aa_set_set_nat_bool(inj_on_set_nat_nat(nat_set_encode),collect_set_nat(finite_finite_nat))) ).

% inj_on_set_encode
tff(fact_10206_ATP_Olambda__1,axiom,
    ! [Uu: option_nat] : aa_option_nat_assn(aTP_Lamm_a,Uu) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,v),aa_nat_nat(suc,zero_zero_nat),vg),vh)),tia)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uu),aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,v),aa_nat_nat(suc,zero_zero_nat),vg),vh)),vi)))) ).

% ATP.lambda_1
tff(fact_10207_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ajg,Uu) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_2
tff(fact_10208_ATP_Olambda__3,axiom,
    ! [Uu: code_integer] : aa_Cod3548095111223497120nteger(aTP_Lamm_yf,Uu) = aa_set6745602309116264572nteger(insert4913895101485356395nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Uu),Uu)),bot_bo4276436098303576167nteger) ).

% ATP.lambda_3
tff(fact_10209_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat7062706319057040595at_nat(aTP_Lamm_yd,Uu) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),Uu)),bot_bo2099793752762293965at_nat) ).

% ATP.lambda_4
tff(fact_10210_ATP_Olambda__5,axiom,
    ! [Uu: int] : aa_int8519581412571446831nt_int(aTP_Lamm_ye,Uu) = aa_set6728540191533482748nt_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Uu),Uu)),bot_bo1796632182523588997nt_int) ).

% ATP.lambda_5
tff(fact_10211_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ajl,Uu) = aa_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,Uu)) ).

% ATP.lambda_6
tff(fact_10212_ATP_Olambda__7,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_akb,Uu))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_7
tff(fact_10213_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_aij,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(power_power_real(zero_zero_real),Uu)) ).

% ATP.lambda_8
tff(fact_10214_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ail,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_9
tff(fact_10215_ATP_Olambda__10,axiom,
    ! [Uu: vEBT_VEBT] : aa_VEB896440568906691887T_VEBT(aTP_Lamm_nk,Uu) = aa_VEB896440568906691887T_VEBT(produc537772716801021591T_VEBT(Uu),Uu) ).

% ATP.lambda_10
tff(fact_10216_ATP_Olambda__11,axiom,
    ! [Uu: code_integer] : aa_Cod6977894654539884864nteger(aTP_Lamm_np,Uu) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Uu),Uu) ).

% ATP.lambda_11
tff(fact_10217_ATP_Olambda__12,axiom,
    ! [Uu: real] : aa_rea5330357063016736847l_real(aTP_Lamm_nl,Uu) = aa_rea5330357063016736847l_real(produc4511245868158468465l_real(Uu),Uu) ).

% ATP.lambda_12
tff(fact_10218_ATP_Olambda__13,axiom,
    ! [Uu: bool] : aa_boo3115168258980966476l_bool(aTP_Lamm_nm,Uu) = aa_boo3115168258980966476l_bool(produc8297744024737662807l_bool(Uu),Uu) ).

% ATP.lambda_13
tff(fact_10219_ATP_Olambda__14,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_nn,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),Uu) ).

% ATP.lambda_14
tff(fact_10220_ATP_Olambda__15,axiom,
    ! [Uu: int] : aa_int7462565939943104719nt_int(aTP_Lamm_no,Uu) = aa_int7462565939943104719nt_int(product_Pair_int_int(Uu),Uu) ).

% ATP.lambda_15
tff(fact_10221_ATP_Olambda__16,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_rb,Uu) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),one_one_complex) ).

% ATP.lambda_16
tff(fact_10222_ATP_Olambda__17,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_rd,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_17
tff(fact_10223_ATP_Olambda__18,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_rc,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ).

% ATP.lambda_18
tff(fact_10224_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_rf,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_19
tff(fact_10225_ATP_Olambda__20,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_re,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_20
tff(fact_10226_ATP_Olambda__21,axiom,
    ! [Uu: vEBT_VEBT] : aa_VEB8154608084236005916T_VEBT(aTP_Lamm_zk,Uu) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Uu),bot_bo8194388402131092736T_VEBT) ).

% ATP.lambda_21
tff(fact_10227_ATP_Olambda__22,axiom,
    ! [Uu: real] : aa_real_set_real(aTP_Lamm_zl,Uu) = aa_set_real_set_real(insert_real(Uu),bot_bot_set_real) ).

% ATP.lambda_22
tff(fact_10228_ATP_Olambda__23,axiom,
    ! [Uu: bool] : aa_bool_set_bool(aTP_Lamm_zm,Uu) = aa_set_bool_set_bool(insert_bool(Uu),bot_bot_set_bool) ).

% ATP.lambda_23
tff(fact_10229_ATP_Olambda__24,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_zo,Uu) = aa_set_nat_set_nat(insert_nat(Uu),bot_bot_set_nat) ).

% ATP.lambda_24
tff(fact_10230_ATP_Olambda__25,axiom,
    ! [Uu: int] : aa_int_set_int(aTP_Lamm_zn,Uu) = aa_set_int_set_int(insert_int(Uu),bot_bot_set_int) ).

% ATP.lambda_25
tff(fact_10231_ATP_Olambda__26,axiom,
    ! [Uu: code_integer] : aa_Cod5401184035482103966nteger(aTP_Lamm_ra,Uu) = insert4913895101485356395nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Uu),Uu)) ).

% ATP.lambda_26
tff(fact_10232_ATP_Olambda__27,axiom,
    ! [Uu: complex] : aa_com7092214031347251183omplex(aTP_Lamm_qx,Uu) = insert3126710022685806477omplex(produc101793102246108661omplex(Uu,Uu)) ).

% ATP.lambda_27
tff(fact_10233_ATP_Olambda__28,axiom,
    ! [Uu: nat] : aa_nat5789291038776893201at_nat(aTP_Lamm_qy,Uu) = insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),Uu)) ).

% ATP.lambda_28
tff(fact_10234_ATP_Olambda__29,axiom,
    ! [Uu: int] : aa_int1478388870204936557nt_int(aTP_Lamm_qz,Uu) = insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Uu),Uu)) ).

% ATP.lambda_29
tff(fact_10235_ATP_Olambda__30,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_akh,Uu) = aa_num_option_num(some_num,aa_num_num(bit1,Uu)) ).

% ATP.lambda_30
tff(fact_10236_ATP_Olambda__31,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_akf,Uu) = aa_num_option_num(some_num,aa_num_num(bit0,Uu)) ).

% ATP.lambda_31
tff(fact_10237_ATP_Olambda__32,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_akt,Uu) = product_Pair_int_int(aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_32
tff(fact_10238_ATP_Olambda__33,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_akw,Uu) = product_Pair_int_int(aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_33
tff(fact_10239_ATP_Olambda__34,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ajv,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_34
tff(fact_10240_ATP_Olambda__35,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_akk,Uu) = aa_num_option_num(some_num,one) ).

% ATP.lambda_35
tff(fact_10241_ATP_Olambda__36,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_akn(Uu),Uua) = case_num_option_num(aa_num_option_num(some_num,one),aTP_Lamm_akl(Uua),aTP_Lamm_akm(Uua),Uu) ).

% ATP.lambda_36
tff(fact_10242_ATP_Olambda__37,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ajm(Uu),Uua) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_37
tff(fact_10243_ATP_Olambda__38,axiom,
    ! [Uu: list_VEBT_VEBTi,Uua: nat] : aa_nat2103294567536003731_VEBTi(aTP_Lamm_gi(Uu),Uua) = if_option_VEBT_VEBTi(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),size_s7982070591426661849_VEBTi(Uu)),some_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uu),Uua)),none_VEBT_VEBTi) ).

% ATP.lambda_38
tff(fact_10244_ATP_Olambda__39,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: nat] : aa_nat3521111894342791148T_VEBT(aTP_Lamm_gj(Uu),Uua) = if_option_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),size_s6755466524823107622T_VEBT(Uu)),some_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uu),Uua)),none_VEBT_VEBT) ).

% ATP.lambda_39
tff(fact_10245_ATP_Olambda__40,axiom,
    ! [Uu: list_P6011104703257516679at_nat,Uua: nat] : aa_nat1535749575751470009at_nat(aTP_Lamm_gk(Uu),Uua) = if_opt6109864365331422477at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),size_s5460976970255530739at_nat(Uu)),aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,nth_Pr7617993195940197384at_nat(Uu,Uua)),none_P5556105721700978146at_nat) ).

% ATP.lambda_40
tff(fact_10246_ATP_Olambda__41,axiom,
    ! [Uu: list_real,Uua: nat] : aa_nat_option_real(aTP_Lamm_gm(Uu),Uua) = if_option_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),aa_list_real_nat(size_size_list_real,Uu)),aa_real_option_real(some_real,aa_nat_real(nth_real(Uu),Uua)),none_real) ).

% ATP.lambda_41
tff(fact_10247_ATP_Olambda__42,axiom,
    ! [Uu: list_bool,Uua: nat] : aa_nat_option_bool(aTP_Lamm_gn(Uu),Uua) = if_option_bool(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),aa_list_bool_nat(size_size_list_bool,Uu)),aa_bool_option_bool(some_bool,aa_nat_bool(nth_bool(Uu),Uua)),none_bool) ).

% ATP.lambda_42
tff(fact_10248_ATP_Olambda__43,axiom,
    ! [Uu: list_num,Uua: nat] : aa_nat_option_num(aTP_Lamm_gl(Uu),Uua) = if_option_num(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),size_size_list_num(Uu)),aa_num_option_num(some_num,aa_nat_num(nth_num(Uu),Uua)),none_num) ).

% ATP.lambda_43
tff(fact_10249_ATP_Olambda__44,axiom,
    ! [Uu: list_nat,Uua: nat] : aa_nat_option_nat(aTP_Lamm_go(Uu),Uua) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),aa_list_nat_nat(size_size_list_nat,Uu)),aa_nat_option_nat(some_nat,aa_nat_nat(nth_nat(Uu),Uua)),none_nat) ).

% ATP.lambda_44
tff(fact_10250_ATP_Olambda__45,axiom,
    ! [Uu: list_int,Uua: nat] : aa_nat_option_int(aTP_Lamm_gp(Uu),Uua) = if_option_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),aa_list_int_nat(size_size_list_int,Uu)),aa_int_option_int(some_int,aa_nat_int(nth_int(Uu),Uua)),none_int) ).

% ATP.lambda_45
tff(fact_10251_ATP_Olambda__46,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_akp,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_46
tff(fact_10252_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_ako,Uu),Uua) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_akn(Uua)),Uu) ).

% ATP.lambda_47
tff(fact_10253_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_akl(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_akf,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_48
tff(fact_10254_ATP_Olambda__49,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_akj(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_akf,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_49
tff(fact_10255_ATP_Olambda__50,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aiz(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(power_power_real(aa_real_real(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_50
tff(fact_10256_ATP_Olambda__51,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ajk(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_51
tff(fact_10257_ATP_Olambda__52,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ajo(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_52
tff(fact_10258_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ajz,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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) = Uu ) ) ) ).

% ATP.lambda_53
tff(fact_10259_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ajy,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(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) = Uu ) ) ) ).

% ATP.lambda_54
tff(fact_10260_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa_int_bool(aa_real_fun_int_bool(aTP_Lamm_akc,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_55
tff(fact_10261_ATP_Olambda__56,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_akd(Uu),Uua))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_56
tff(fact_10262_ATP_Olambda__57,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aTP_Lamm_ob(Uu),Uua))
    <=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uua)),Uu))
        & distinct_VEBT_VEBT(Uua) ) ) ).

% ATP.lambda_57
tff(fact_10263_ATP_Olambda__58,axiom,
    ! [Uu: set_Code_integer,Uua: list_Code_integer] :
      ( pp(aa_lis9036377640098331692r_bool(aTP_Lamm_og(Uu),Uua))
    <=> ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_Code_integer2(Uua)),Uu))
        & distin1543349897113766820nteger(Uua) ) ) ).

% ATP.lambda_58
tff(fact_10264_ATP_Olambda__59,axiom,
    ! [Uu: set_complex,Uua: list_complex] :
      ( pp(aa_list_complex_bool(aTP_Lamm_of(Uu),Uua))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uua)),Uu))
        & distinct_complex(Uua) ) ) ).

% ATP.lambda_59
tff(fact_10265_ATP_Olambda__60,axiom,
    ! [Uu: set_real,Uua: list_real] :
      ( pp(aa_list_real_bool(aTP_Lamm_oc(Uu),Uua))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Uua)),Uu))
        & pp(aa_list_real_bool(distinct_real,Uua)) ) ) ).

% ATP.lambda_60
tff(fact_10266_ATP_Olambda__61,axiom,
    ! [Uu: set_bool,Uua: list_bool] :
      ( pp(aa_list_bool_bool(aTP_Lamm_od(Uu),Uua))
    <=> ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Uua)),Uu))
        & pp(aa_list_bool_bool(distinct_bool,Uua)) ) ) ).

% ATP.lambda_61
tff(fact_10267_ATP_Olambda__62,axiom,
    ! [Uu: set_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_oe(Uu),Uua))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Uua)),Uu))
        & pp(aa_list_nat_bool(distinct_nat,Uua)) ) ) ).

% ATP.lambda_62
tff(fact_10268_ATP_Olambda__63,axiom,
    ! [Uu: set_int,Uua: list_int] :
      ( pp(aa_list_int_bool(aTP_Lamm_oh(Uu),Uua))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Uua)),Uu))
        & pp(aa_list_int_bool(distinct_int,Uua)) ) ) ).

% ATP.lambda_63
tff(fact_10269_ATP_Olambda__64,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aTP_Lamm_nx(Uu),Uua))
    <=> ( ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uua) = Uu )
        & distinct_VEBT_VEBT(Uua) ) ) ).

% ATP.lambda_64
tff(fact_10270_ATP_Olambda__65,axiom,
    ! [Uu: set_real,Uua: list_real] :
      ( pp(aa_list_real_bool(aTP_Lamm_nz(Uu),Uua))
    <=> ( ( aa_lis3769085826154824284t_real(set_real2,Uua) = Uu )
        & pp(aa_list_real_bool(distinct_real,Uua)) ) ) ).

% ATP.lambda_65
tff(fact_10271_ATP_Olambda__66,axiom,
    ! [Uu: set_bool,Uua: list_bool] :
      ( pp(aa_list_bool_bool(aTP_Lamm_oa(Uu),Uua))
    <=> ( ( aa_lis3587183946281582166t_bool(set_bool2,Uua) = Uu )
        & pp(aa_list_bool_bool(distinct_bool,Uua)) ) ) ).

% ATP.lambda_66
tff(fact_10272_ATP_Olambda__67,axiom,
    ! [Uu: set_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ny(Uu),Uua))
    <=> ( ( aa_list_nat_set_nat(set_nat2,Uua) = Uu )
        & pp(aa_list_nat_bool(distinct_nat,Uua)) ) ) ).

% ATP.lambda_67
tff(fact_10273_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aja(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_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(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_68
tff(fact_10274_ATP_Olambda__69,axiom,
    ! [Uu: set_op4508134149509766951at_nat,Uua: option4927543243414619207at_nat] :
      ( pp(aa_opt6800557945374683724t_bool(aTP_Lamm_abw(Uu),Uua))
    <=> ( member3954567711264315760at_nat(Uua,Uu)
        & ( Uua != none_P5556105721700978146at_nat ) ) ) ).

% ATP.lambda_69
tff(fact_10275_ATP_Olambda__70,axiom,
    ! [Uu: set_option_num,Uua: option_num] :
      ( pp(aa_option_num_bool(aTP_Lamm_aby(Uu),Uua))
    <=> ( member_option_num(Uua,Uu)
        & ( Uua != none_num ) ) ) ).

% ATP.lambda_70
tff(fact_10276_ATP_Olambda__71,axiom,
    ! [Uu: set_option_nat,Uua: option_nat] :
      ( pp(aa_option_nat_bool(aTP_Lamm_abx(Uu),Uua))
    <=> ( member_option_nat(Uua,Uu)
        & ( Uua != none_nat ) ) ) ).

% ATP.lambda_71
tff(fact_10277_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ait,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_72
tff(fact_10278_ATP_Olambda__73,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aik(Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(power_power_real(Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_73
tff(fact_10279_ATP_Olambda__74,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(aTP_Lamm_aco(Uu),Uua))
    <=> ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(Uua),Uu))
        & pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,Uua)) ) ) ).

% ATP.lambda_74
tff(fact_10280_ATP_Olambda__75,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( pp(aa_set_complex_bool(aTP_Lamm_acn(Uu),Uua))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(Uua),Uu))
        & pp(aa_set_complex_bool(finite3207457112153483333omplex,Uua)) ) ) ).

% ATP.lambda_75
tff(fact_10281_ATP_Olambda__76,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_acm,Uu),Uua))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Uua),Uu))
        & pp(aa_set_nat_bool(finite_finite_nat,Uua)) ) ) ).

% ATP.lambda_76
tff(fact_10282_ATP_Olambda__77,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aTP_Lamm_acp(Uu),Uua))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(Uua),Uu))
        & pp(aa_set_int_bool(finite_finite_int,Uua)) ) ) ).

% ATP.lambda_77
tff(fact_10283_ATP_Olambda__78,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aip,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_78
tff(fact_10284_ATP_Olambda__79,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aio,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_79
tff(fact_10285_ATP_Olambda__80,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat2161576163539049207BT_nat(aTP_Lamm_ww(Uu),Uua) = aa_set1503868022174493200BT_nat(insert8978894354669351395BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Uu),Uua)),bot_bo1642239108664514429BT_nat) ).

% ATP.lambda_80
tff(fact_10286_ATP_Olambda__81,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod3548095111223497120nteger(aTP_Lamm_xa(Uu),Uua) = aa_set6745602309116264572nteger(insert4913895101485356395nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Uu),Uua)),bot_bo4276436098303576167nteger) ).

% ATP.lambda_81
tff(fact_10287_ATP_Olambda__82,axiom,
    ! [Uu: nat,Uua: num] : aa_num4662830302079825491at_num(aTP_Lamm_xc(Uu),Uua) = aa_set8037964202635646076at_num(insert8920054152555992091at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(Uu),Uua)),bot_bo7038385379056416535at_num) ).

% ATP.lambda_82
tff(fact_10288_ATP_Olambda__83,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7062706319057040595at_nat(aTP_Lamm_wu(Uu),Uua) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),Uua)),bot_bo2099793752762293965at_nat) ).

% ATP.lambda_83
tff(fact_10289_ATP_Olambda__84,axiom,
    ! [Uu: int,Uua: int] : aa_int8519581412571446831nt_int(aTP_Lamm_wy(Uu),Uua) = aa_set6728540191533482748nt_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Uu),Uua)),bot_bo1796632182523588997nt_int) ).

% ATP.lambda_84
tff(fact_10290_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_acs,Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_85
tff(fact_10291_ATP_Olambda__86,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_acr,Uu),Uua))
    <=> ( aa_nat_real(power_power_real(Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_86
tff(fact_10292_ATP_Olambda__87,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aiy(Uu),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(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_87
tff(fact_10293_ATP_Olambda__88,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ake,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi))
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_88
tff(fact_10294_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_aek(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_89
tff(fact_10295_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ael(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(power_power_real(zero_zero_real),Uua)) ).

% ATP.lambda_90
tff(fact_10296_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_aej(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_91
tff(fact_10297_ATP_Olambda__92,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: vEBT_VEBT] : aa_VEB6922606862176952244t_real(aTP_Lamm_aee(Uu),Uua) = cons_real(aa_VEBT_VEBT_real(Uu,Uua),nil_real) ).

% ATP.lambda_92
tff(fact_10298_ATP_Olambda__93,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: vEBT_VEBT] : aa_VEB4395186035530729816st_nat(aTP_Lamm_aeh(Uu),Uua) = aa_list_nat_list_nat(cons_nat(aa_VEBT_VEBT_nat(Uu,Uua)),nil_nat) ).

% ATP.lambda_93
tff(fact_10299_ATP_Olambda__94,axiom,
    ! [Uu: fun_Pr4298194452408323251l_bool,Uua: produc2804947525362035943l_bool] : aa_Pro3826829133813349938t_bool(aTP_Lamm_aef(Uu),Uua) = cons_bool(aa_Pro9114106320147817132l_bool(Uu,Uua),nil_bool) ).

% ATP.lambda_94
tff(fact_10300_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] : aa_nat_list_bool(aTP_Lamm_aeg(Uu),Uua) = cons_bool(aa_nat_bool(Uu,Uua),nil_bool) ).

% ATP.lambda_95
tff(fact_10301_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_list_nat(aTP_Lamm_aei(Uu),Uua) = aa_list_nat_list_nat(cons_nat(aa_nat_nat(Uu,Uua)),nil_nat) ).

% ATP.lambda_96
tff(fact_10302_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_set_real(aTP_Lamm_abs(Uu),Uua) = aa_set_real_set_real(insert_real(aa_nat_real(Uu,Uua)),bot_bot_set_real) ).

% ATP.lambda_97
tff(fact_10303_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_set_nat,Uua: nat] : aa_nat_set_set_nat(aTP_Lamm_abr(Uu),Uua) = aa_set1013257548857649334et_nat(insert_set_nat(aa_nat_set_nat(Uu,Uua)),bot_bot_set_set_nat) ).

% ATP.lambda_98
tff(fact_10304_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abv(Uu),Uua) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(Uu,Uua)),bot_bot_set_nat) ).

% ATP.lambda_99
tff(fact_10305_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_set_int(aTP_Lamm_abt(Uu),Uua) = aa_set_int_set_int(insert_int(aa_nat_int(Uu,Uua)),bot_bot_set_int) ).

% ATP.lambda_100
tff(fact_10306_ATP_Olambda__101,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_set_int(aTP_Lamm_abu(Uu),Uua) = aa_set_int_set_int(insert_int(aa_int_int(Uu,Uua)),bot_bot_set_int) ).

% ATP.lambda_101
tff(fact_10307_ATP_Olambda__102,axiom,
    ! [Uu: list_int,Uua: list_int] : aa_lis8041128603315886278st_int(aTP_Lamm_qw(Uu),Uua) = map_int_list_int(aTP_Lamm_qv(Uua),Uu) ).

% ATP.lambda_102
tff(fact_10308_ATP_Olambda__103,axiom,
    ! [Uu: list_nat,Uua: list_nat] : aa_lis99722578464457230st_nat(aTP_Lamm_qu(Uu),Uua) = map_nat_list_nat(aTP_Lamm_qt(Uua),Uu) ).

% ATP.lambda_103
tff(fact_10309_ATP_Olambda__104,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ajt,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).

% ATP.lambda_104
tff(fact_10310_ATP_Olambda__105,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_akg(Uu),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_105
tff(fact_10311_ATP_Olambda__106,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_aji(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_106
tff(fact_10312_ATP_Olambda__107,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ajh(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_107
tff(fact_10313_ATP_Olambda__108,axiom,
    ! [Uu: list_list_nat,Uua: nat] : aa_nat_list_list_nat(aTP_Lamm_qr(Uu),Uua) = map_li7225945977422193158st_nat(cons_nat(Uua),product_lists_nat(Uu)) ).

% ATP.lambda_108
tff(fact_10314_ATP_Olambda__109,axiom,
    ! [Uu: list_list_int,Uua: int] : aa_int_list_list_int(aTP_Lamm_qs(Uu),Uua) = map_li4896172289311737022st_int(cons_int(Uua),product_lists_int(Uu)) ).

% ATP.lambda_109
tff(fact_10315_ATP_Olambda__110,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: vEBT_VEBT] : aa_VEB1879592818988569141T_VEBT(aTP_Lamm_iu(Uu),Uua) = map_VE1720758354293053111T_VEBT(produc537772716801021591T_VEBT(Uua),Uu) ).

% ATP.lambda_110
tff(fact_10316_ATP_Olambda__111,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: real] : aa_rea4792338944335410167T_VEBT(aTP_Lamm_ja(Uu),Uua) = map_VE1210512188453230445T_VEBT(produc6931449550656315951T_VEBT(Uua),Uu) ).

% ATP.lambda_111
tff(fact_10317_ATP_Olambda__112,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: nat] : aa_nat5381440928738364023T_VEBT(aTP_Lamm_ix(Uu),Uua) = map_VE5619196432070217417T_VEBT(produc599794634098209291T_VEBT(Uua),Uu) ).

% ATP.lambda_112
tff(fact_10318_ATP_Olambda__113,axiom,
    ! [Uu: list_Code_integer,Uua: code_integer] : aa_Cod4629117397528844102nteger(aTP_Lamm_lc(Uu),Uua) = map_Co3589949550033412536nteger(produc1086072967326762835nteger(Uua),Uu) ).

% ATP.lambda_113
tff(fact_10319_ATP_Olambda__114,axiom,
    ! [Uu: list_real,Uua: vEBT_VEBT] : aa_VEB6766512809318060311T_real(aTP_Lamm_iv(Uu),Uua) = map_re8618229306769252225T_real(produc8117437818029410057T_real(Uua),Uu) ).

% ATP.lambda_114
tff(fact_10320_ATP_Olambda__115,axiom,
    ! [Uu: list_real,Uua: real] : aa_rea7790490538786641877l_real(aTP_Lamm_jc(Uu),Uua) = map_re3533924840734919879l_real(produc4511245868158468465l_real(Uua),Uu) ).

% ATP.lambda_115
tff(fact_10321_ATP_Olambda__116,axiom,
    ! [Uu: list_real,Uua: nat] : aa_nat3009726972993872213t_real(aTP_Lamm_iy(Uu),Uua) = map_re7303296592222247787t_real(produc7837566107596912789t_real(Uua),Uu) ).

% ATP.lambda_116
tff(fact_10322_ATP_Olambda__117,axiom,
    ! [Uu: list_bool,Uua: vEBT_VEBT] : aa_VEB6401713452465299736T_bool(aTP_Lamm_iw(Uu),Uua) = map_bo5103901678433843959T_bool(produc3035007008238225530T_bool(Uua),Uu) ).

% ATP.lambda_117
tff(fact_10323_ATP_Olambda__118,axiom,
    ! [Uu: list_bool,Uua: real] : aa_rea983618398234757850l_bool(aTP_Lamm_jd(Uu),Uua) = map_bo2606389728334092083l_bool(produc3350604258665657362l_bool(Uua),Uu) ).

% ATP.lambda_118
tff(fact_10324_ATP_Olambda__119,axiom,
    ! [Uu: list_bool,Uua: nat] : aa_nat3396843857113850202t_bool(aTP_Lamm_iz(Uu),Uua) = map_bo21327563268898831t_bool(produc3913204875133826030t_bool(Uua),Uu) ).

% ATP.lambda_119
tff(fact_10325_ATP_Olambda__120,axiom,
    ! [Uu: list_num,Uua: nat] : aa_nat1484155492932414531at_num(aTP_Lamm_ld(Uu),Uua) = map_nu4721551698833171051at_num(product_Pair_nat_num(Uua),Uu) ).

% ATP.lambda_120
tff(fact_10326_ATP_Olambda__121,axiom,
    ! [Uu: list_nat,Uua: vEBT_VEBT] : aa_VEB6663915520319177915BT_nat(aTP_Lamm_la(Uu),Uua) = map_na4631810538828370761BT_nat(produc738532404422230701BT_nat(Uua),Uu) ).

% ATP.lambda_121
tff(fact_10327_ATP_Olambda__122,axiom,
    ! [Uu: list_nat,Uua: real] : aa_rea752208095135664761al_nat(aTP_Lamm_jb(Uu),Uua) = map_na1309146153819616583al_nat(produc3181502643871035669al_nat(Uua),Uu) ).

% ATP.lambda_122
tff(fact_10328_ATP_Olambda__123,axiom,
    ! [Uu: list_nat,Uua: nat] : aa_nat5768935903493067769at_nat(aTP_Lamm_kz(Uu),Uua) = map_na7298421622053143531at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_123
tff(fact_10329_ATP_Olambda__124,axiom,
    ! [Uu: list_int,Uua: int] : aa_int8473326307649047381nt_int(aTP_Lamm_lb(Uu),Uua) = map_in7157766398909135175nt_int(product_Pair_int_int(Uua),Uu) ).

% ATP.lambda_124
tff(fact_10330_ATP_Olambda__125,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_aka,Uu),Uua))
    <=> ( aa_real_real(exp_real,Uua) = Uu ) ) ).

% ATP.lambda_125
tff(fact_10331_ATP_Olambda__126,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiv,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_126
tff(fact_10332_ATP_Olambda__127,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aju,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int)))) ).

% ATP.lambda_127
tff(fact_10333_ATP_Olambda__128,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_alk(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(Uu),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_128
tff(fact_10334_ATP_Olambda__129,axiom,
    ! [Uu: list_set_real,Uua: set_real] :
      ( pp(aa_set_real_bool(aTP_Lamm_xp(Uu),Uua))
    <=> member_set_real(Uua,set_set_real2(Uu)) ) ).

% ATP.lambda_129
tff(fact_10335_ATP_Olambda__130,axiom,
    ! [Uu: list_set_bool,Uua: set_bool] :
      ( pp(aa_set_bool_bool(aTP_Lamm_xq(Uu),Uua))
    <=> member_set_bool(Uua,set_set_bool2(Uu)) ) ).

% ATP.lambda_130
tff(fact_10336_ATP_Olambda__131,axiom,
    ! [Uu: list_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_xs(Uu),Uua))
    <=> pp(aa_set_set_nat_bool(member_set_nat(Uua),set_set_nat2(Uu))) ) ).

% ATP.lambda_131
tff(fact_10337_ATP_Olambda__132,axiom,
    ! [Uu: list_set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aTP_Lamm_xr(Uu),Uua))
    <=> member_set_int(Uua,set_set_int2(Uu)) ) ).

% ATP.lambda_132
tff(fact_10338_ATP_Olambda__133,axiom,
    ! [Uu: assn,Uua: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(aTP_Lamm_sk,Uu),Uua))
    <=> ord_less_eq_assn(Uua,Uu) ) ).

% ATP.lambda_133
tff(fact_10339_ATP_Olambda__134,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(aTP_Lamm_cj(Uu),Uua))
    <=> pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(Uua),Uu)) ) ).

% ATP.lambda_134
tff(fact_10340_ATP_Olambda__135,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( pp(aa_set_complex_bool(aTP_Lamm_ci(Uu),Uua))
    <=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(Uua),Uu)) ) ).

% ATP.lambda_135
tff(fact_10341_ATP_Olambda__136,axiom,
    ! [Uu: set_real,Uua: set_real] :
      ( pp(aa_set_real_bool(aa_set4823678304375637230l_bool(aTP_Lamm_sm,Uu),Uua))
    <=> pp(aa_set_real_bool(ord_less_eq_set_real(Uua),Uu)) ) ).

% ATP.lambda_136
tff(fact_10342_ATP_Olambda__137,axiom,
    ! [Uu: set_bool,Uua: set_bool] :
      ( pp(aa_set_bool_bool(aa_set36795378406917470l_bool(aTP_Lamm_so,Uu),Uua))
    <=> pp(aa_set_bool_bool(ord_less_eq_set_bool(Uua),Uu)) ) ).

% ATP.lambda_137
tff(fact_10343_ATP_Olambda__138,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_ch,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,Uua),Uu)) ) ).

% ATP.lambda_138
tff(fact_10344_ATP_Olambda__139,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aa_set7694327796961020398t_bool(aTP_Lamm_ck,Uu),Uua))
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(Uua),Uu)) ) ).

% ATP.lambda_139
tff(fact_10345_ATP_Olambda__140,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_abz,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu)) ) ).

% ATP.lambda_140
tff(fact_10346_ATP_Olambda__141,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_aca,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Uua),Uu)) ) ).

% ATP.lambda_141
tff(fact_10347_ATP_Olambda__142,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cf,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_142
tff(fact_10348_ATP_Olambda__143,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_acb,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).

% ATP.lambda_143
tff(fact_10349_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_alj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_144
tff(fact_10350_ATP_Olambda__145,axiom,
    ! [Uu: assn,Uua: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(aTP_Lamm_sl,Uu),Uua))
    <=> ord_less_assn(Uua,Uu) ) ).

% ATP.lambda_145
tff(fact_10351_ATP_Olambda__146,axiom,
    ! [Uu: set_real,Uua: set_real] :
      ( pp(aa_set_real_bool(aa_set4823678304375637230l_bool(aTP_Lamm_sn,Uu),Uua))
    <=> pp(aa_set_real_bool(ord_less_set_real(Uua),Uu)) ) ).

% ATP.lambda_146
tff(fact_10352_ATP_Olambda__147,axiom,
    ! [Uu: set_bool,Uua: set_bool] :
      ( pp(aa_set_bool_bool(aa_set36795378406917470l_bool(aTP_Lamm_sp,Uu),Uua))
    <=> pp(aa_set_bool_bool(ord_less_set_bool(Uua),Uu)) ) ).

% ATP.lambda_147
tff(fact_10353_ATP_Olambda__148,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_sq,Uu),Uua))
    <=> pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_set_nat,Uua),Uu)) ) ).

% ATP.lambda_148
tff(fact_10354_ATP_Olambda__149,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( pp(aa_set_int_bool(aa_set7694327796961020398t_bool(aTP_Lamm_sr,Uu),Uua))
    <=> pp(aa_set_int_bool(ord_less_set_int(Uua),Uu)) ) ).

% ATP.lambda_149
tff(fact_10355_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cg,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_150
tff(fact_10356_ATP_Olambda__151,axiom,
    ! [Uu: complex,Uua: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ta,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uua),Uu) ).

% ATP.lambda_151
tff(fact_10357_ATP_Olambda__152,axiom,
    ! [Uu: assn,Uua: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(aTP_Lamm_sz,Uu),Uua) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),Uu) ).

% ATP.lambda_152
tff(fact_10358_ATP_Olambda__153,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_sv(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu) ).

% ATP.lambda_153
tff(fact_10359_ATP_Olambda__154,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_sw,Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Uua),Uu) ).

% ATP.lambda_154
tff(fact_10360_ATP_Olambda__155,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sx,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_155
tff(fact_10361_ATP_Olambda__156,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_sy,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).

% ATP.lambda_156
tff(fact_10362_ATP_Olambda__157,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_vz(Uu),Uua) = aa_real_real(minus_minus_real(Uua),Uu) ).

% ATP.lambda_157
tff(fact_10363_ATP_Olambda__158,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_wa,Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uua),Uu) ).

% ATP.lambda_158
tff(fact_10364_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_159
tff(fact_10365_ATP_Olambda__160,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_wb,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uua),Uu) ).

% ATP.lambda_160
tff(fact_10366_ATP_Olambda__161,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_tg(Uu),Uua) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uua),Uu) ).

% ATP.lambda_161
tff(fact_10367_ATP_Olambda__162,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_tc(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu) ).

% ATP.lambda_162
tff(fact_10368_ATP_Olambda__163,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aa_rat_fun_rat_rat(aTP_Lamm_td,Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uua),Uu) ).

% ATP.lambda_163
tff(fact_10369_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_te,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_164
tff(fact_10370_ATP_Olambda__165,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_tf,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_165
tff(fact_10371_ATP_Olambda__166,axiom,
    ! [Uu: list_VEBT_VEBTi,Uua: array_VEBT_VEBTi] : aa_arr8637723718858317150i_assn(aa_lis341327733535136106i_assn(aTP_Lamm_os,Uu),Uua) = snga_assn_VEBT_VEBTi(Uua,Uu) ).

% ATP.lambda_166
tff(fact_10372_ATP_Olambda__167,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ajb,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_167
tff(fact_10373_ATP_Olambda__168,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ajf,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_168
tff(fact_10374_ATP_Olambda__169,axiom,
    ! [Uu: vEBT_VEBT,Uua: real] : aa_rea2776067176874379751T_VEBT(aTP_Lamm_vh(Uu),Uua) = aa_VEB1592731369666541301T_VEBT(produc6931449550656315951T_VEBT(Uua),Uu) ).

% ATP.lambda_169
tff(fact_10375_ATP_Olambda__170,axiom,
    ! [Uu: vEBT_VEBT,Uua: bool] : aa_boo4381339273886475241T_VEBT(aTP_Lamm_vj(Uu),Uua) = produc9074556814250467700T_VEBT(Uua,Uu) ).

% ATP.lambda_170
tff(fact_10376_ATP_Olambda__171,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat3316634782163766759T_VEBT(aTP_Lamm_vl(Uu),Uua) = aa_VEB2495835560080880209T_VEBT(produc599794634098209291T_VEBT(Uua),Uu) ).

% ATP.lambda_171
tff(fact_10377_ATP_Olambda__172,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aTP_Lamm_vg(Uu),Uua) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Uua),Uu) ).

% ATP.lambda_172
tff(fact_10378_ATP_Olambda__173,axiom,
    ! [Uu: bool,Uua: real] : aa_rea3414591235091353162l_bool(aTP_Lamm_vi(Uu),Uua) = aa_boo8879657058893838011l_bool(produc3350604258665657362l_bool(Uua),Uu) ).

% ATP.lambda_173
tff(fact_10379_ATP_Olambda__174,axiom,
    ! [Uu: bool,Uua: bool] : aa_boo3115168258980966476l_bool(aTP_Lamm_vk(Uu),Uua) = aa_boo3115168258980966476l_bool(produc8297744024737662807l_bool(Uua),Uu) ).

% ATP.lambda_174
tff(fact_10380_ATP_Olambda__175,axiom,
    ! [Uu: bool,Uua: nat] : aa_nat9043171019412722250t_bool(aTP_Lamm_vm(Uu),Uua) = aa_boo8466755144949572503t_bool(produc3913204875133826030t_bool(Uua),Uu) ).

% ATP.lambda_175
tff(fact_10381_ATP_Olambda__176,axiom,
    ! [Uu: num,Uua: nat] : aa_nat8311465100035950781at_num(aTP_Lamm_vo(Uu),Uua) = aa_num5026351239867140595at_num(product_Pair_nat_num(Uua),Uu) ).

% ATP.lambda_176
tff(fact_10382_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEB3542760477308195371BT_nat(aTP_Lamm_vf(Uu),Uua) = aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Uua),Uu) ).

% ATP.lambda_177
tff(fact_10383_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_vn(Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_178
tff(fact_10384_ATP_Olambda__179,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aTP_Lamm_wt(Uu),Uua) = aa_int7462565939943104719nt_int(product_Pair_int_int(Uua),Uu) ).

% ATP.lambda_179
tff(fact_10385_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ain,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_180
tff(fact_10386_ATP_Olambda__181,axiom,
    ! [Uu: list_nat,Uua: nat] : aa_nat_list_nat(aTP_Lamm_qt(Uu),Uua) = aa_list_nat_list_nat(cons_nat(Uua),Uu) ).

% ATP.lambda_181
tff(fact_10387_ATP_Olambda__182,axiom,
    ! [Uu: list_int,Uua: int] : aa_int_list_int(aTP_Lamm_qv(Uu),Uua) = aa_list_int_list_int(cons_int(Uua),Uu) ).

% ATP.lambda_182
tff(fact_10388_ATP_Olambda__183,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_an,Uu),Uua))
    <=> pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uua),Uu)) ) ).

% ATP.lambda_183
tff(fact_10389_ATP_Olambda__184,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ap(Uu),Uua))
    <=> pp(aa_set_complex_bool(member_complex(Uua),Uu)) ) ).

% ATP.lambda_184
tff(fact_10390_ATP_Olambda__185,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ao,Uu),Uua))
    <=> pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_185
tff(fact_10391_ATP_Olambda__186,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_aq(Uu),Uua))
    <=> pp(aa_set_list_nat_bool(member_list_nat(Uua),Uu)) ) ).

% ATP.lambda_186
tff(fact_10392_ATP_Olambda__187,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_ar,Uu),Uua))
    <=> pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_187
tff(fact_10393_ATP_Olambda__188,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_as,Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_188
tff(fact_10394_ATP_Olambda__189,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_at,Uu),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_189
tff(fact_10395_ATP_Olambda__190,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aTP_Lamm_je,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_190
tff(fact_10396_ATP_Olambda__191,axiom,
    ! [Uu: complex,Uua: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_jf,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_191
tff(fact_10397_ATP_Olambda__192,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ji,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_192
tff(fact_10398_ATP_Olambda__193,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_jg,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_193
tff(fact_10399_ATP_Olambda__194,axiom,
    ! [Uu: bool,Uua: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_jj,Uu),Uua))
    <=> ( pp(Uua)
      <=> pp(Uu) ) ) ).

% ATP.lambda_194
tff(fact_10400_ATP_Olambda__195,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_jh,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_195
tff(fact_10401_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_jk,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_196
tff(fact_10402_ATP_Olambda__197,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jl,Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_197
tff(fact_10403_ATP_Olambda__198,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aTP_Lamm_bx(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Uu,Uua))) ) ).

% ATP.lambda_198
tff(fact_10404_ATP_Olambda__199,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_bv(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(Uu,Uua))) ) ).

% ATP.lambda_199
tff(fact_10405_ATP_Olambda__200,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_bn(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_list_nat_nat(Uu,Uua))) ) ).

% ATP.lambda_200
tff(fact_10406_ATP_Olambda__201,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_bp(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(Uu,Uua))) ) ).

% ATP.lambda_201
tff(fact_10407_ATP_Olambda__202,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_br(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu,Uua))) ) ).

% ATP.lambda_202
tff(fact_10408_ATP_Olambda__203,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_bt(Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(Uu,Uua))) ) ).

% ATP.lambda_203
tff(fact_10409_ATP_Olambda__204,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aje(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_204
tff(fact_10410_ATP_Olambda__205,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ajd(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_205
tff(fact_10411_ATP_Olambda__206,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_all(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_206
tff(fact_10412_ATP_Olambda__207,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_akm(Uu),Uua) = aa_num_option_num(some_num,case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_207
tff(fact_10413_ATP_Olambda__208,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_aki(Uu),Uua) = aa_num_option_num(some_num,case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_208
tff(fact_10414_ATP_Olambda__209,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ajp,Uu),Uua))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_209
tff(fact_10415_ATP_Olambda__210,axiom,
    ! [Uu: set_set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_aaf(Uu),Uua))
    <=> pp(aa_set_bool_bool(comple6445916836919887748p_bool,image_5803757493061269640T_bool(member_VEBT_VEBT(Uua),Uu))) ) ).

% ATP.lambda_210
tff(fact_10416_ATP_Olambda__211,axiom,
    ! [Uu: set_set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_aah(Uu),Uua))
    <=> pp(aa_set_bool_bool(comple6445916836919887748p_bool,image_214517817511200108x_bool(member_complex(Uua),Uu))) ) ).

% ATP.lambda_211
tff(fact_10417_ATP_Olambda__212,axiom,
    ! [Uu: set_set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aag(Uu),Uua))
    <=> pp(aa_set_bool_bool(comple6445916836919887748p_bool,image_set_real_bool(member_real(Uua),Uu))) ) ).

% ATP.lambda_212
tff(fact_10418_ATP_Olambda__213,axiom,
    ! [Uu: set_set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_aai(Uu),Uua))
    <=> pp(aa_set_bool_bool(comple6445916836919887748p_bool,image_6778563931047677754t_bool(member_list_nat(Uua),Uu))) ) ).

% ATP.lambda_213
tff(fact_10419_ATP_Olambda__214,axiom,
    ! [Uu: set_set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_aaj(Uu),Uua))
    <=> pp(aa_set_bool_bool(comple6445916836919887748p_bool,image_4022858629048953492t_bool(member_set_nat(Uua),Uu))) ) ).

% ATP.lambda_214
tff(fact_10420_ATP_Olambda__215,axiom,
    ! [Uu: set_set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_aal(Uu),Uua))
    <=> pp(aa_set_bool_bool(comple6445916836919887748p_bool,image_set_nat_bool(member_nat(Uua),Uu))) ) ).

% ATP.lambda_215
tff(fact_10421_ATP_Olambda__216,axiom,
    ! [Uu: set_set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_aak(Uu),Uua))
    <=> pp(aa_set_bool_bool(comple6445916836919887748p_bool,image_set_int_bool(member_int(Uua),Uu))) ) ).

% ATP.lambda_216
tff(fact_10422_ATP_Olambda__217,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_rh(Uu),Uua) = aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_217
tff(fact_10423_ATP_Olambda__218,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ri(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_218
tff(fact_10424_ATP_Olambda__219,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_rg(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_219
tff(fact_10425_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_rk(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_220
tff(fact_10426_ATP_Olambda__221,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_rj(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_221
tff(fact_10427_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_nat] : aa_set_nat_set_nat(aTP_Lamm_abi(Uu),Uua) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(Uu),Uua)) ).

% ATP.lambda_222
tff(fact_10428_ATP_Olambda__223,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat1592460887112715357BT_nat(aTP_Lamm_wx(Uu),Uua) = insert8978894354669351395BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Uu),Uua)) ).

% ATP.lambda_223
tff(fact_10429_ATP_Olambda__224,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod5401184035482103966nteger(aTP_Lamm_xb(Uu),Uua) = insert4913895101485356395nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Uu),Uua)) ).

% ATP.lambda_224
tff(fact_10430_ATP_Olambda__225,axiom,
    ! [Uu: nat,Uua: num] : aa_num2946070018604694747at_num(aTP_Lamm_xd(Uu),Uua) = insert8920054152555992091at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(Uu),Uua)) ).

% ATP.lambda_225
tff(fact_10431_ATP_Olambda__226,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat5789291038776893201at_nat(aTP_Lamm_wv(Uu),Uua) = insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),Uua)) ).

% ATP.lambda_226
tff(fact_10432_ATP_Olambda__227,axiom,
    ! [Uu: int,Uua: int] : aa_int1478388870204936557nt_int(aTP_Lamm_wz(Uu),Uua) = insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Uu),Uua)) ).

% ATP.lambda_227
tff(fact_10433_ATP_Olambda__228,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_alh,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_228
tff(fact_10434_ATP_Olambda__229,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ali,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_229
tff(fact_10435_ATP_Olambda__230,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_akr,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_230
tff(fact_10436_ATP_Olambda__231,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aks,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_231
tff(fact_10437_ATP_Olambda__232,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_wh(Uu),Uua))
    <=> ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uua),Uu)) ) ).

% ATP.lambda_232
tff(fact_10438_ATP_Olambda__233,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_wj(Uu),Uua))
    <=> ~ pp(aa_set_complex_bool(member_complex(Uua),Uu)) ) ).

% ATP.lambda_233
tff(fact_10439_ATP_Olambda__234,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_wi(Uu),Uua))
    <=> ~ pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).

% ATP.lambda_234
tff(fact_10440_ATP_Olambda__235,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_wk(Uu),Uua))
    <=> ~ pp(aa_set_list_nat_bool(member_list_nat(Uua),Uu)) ) ).

% ATP.lambda_235
tff(fact_10441_ATP_Olambda__236,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_wl(Uu),Uua))
    <=> ~ pp(aa_set_set_nat_bool(member_set_nat(Uua),Uu)) ) ).

% ATP.lambda_236
tff(fact_10442_ATP_Olambda__237,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_wm(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_237
tff(fact_10443_ATP_Olambda__238,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_wn(Uu),Uua))
    <=> ~ pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_238
tff(fact_10444_ATP_Olambda__239,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_pu(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(Uu,Uua)) ).

% ATP.lambda_239
tff(fact_10445_ATP_Olambda__240,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_ade(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_240
tff(fact_10446_ATP_Olambda__241,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_pv(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_241
tff(fact_10447_ATP_Olambda__242,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_adh(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_242
tff(fact_10448_ATP_Olambda__243,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_adg(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_243
tff(fact_10449_ATP_Olambda__244,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_adf(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_244
tff(fact_10450_ATP_Olambda__245,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_complex(aTP_Lamm_afx(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_245
tff(fact_10451_ATP_Olambda__246,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_afy(Uu),Uua) = aa_real_real(uminus_uminus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_246
tff(fact_10452_ATP_Olambda__247,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_afz(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_247
tff(fact_10453_ATP_Olambda__248,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_adi(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_248
tff(fact_10454_ATP_Olambda__249,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_adj(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_249
tff(fact_10455_ATP_Olambda__250,axiom,
    ! [Uu: fun_real_set_nat,Uua: real] : aa_rea5080859985761801024et_nat(aTP_Lamm_xl(Uu),Uua) = aa_set6119337032432945050et_nat(some_set_nat,aa_real_set_nat(Uu,Uua)) ).

% ATP.lambda_250
tff(fact_10456_ATP_Olambda__251,axiom,
    ! [Uu: fun_bool_set_nat,Uua: bool] : aa_boo2874362534103293361et_nat(aTP_Lamm_xm(Uu),Uua) = aa_set6119337032432945050et_nat(some_set_nat,aa_bool_set_nat(Uu,Uua)) ).

% ATP.lambda_251
tff(fact_10457_ATP_Olambda__252,axiom,
    ! [Uu: fun_nat_set_nat,Uua: nat] : aa_nat7838062387752323300et_nat(aTP_Lamm_xn(Uu),Uua) = aa_set6119337032432945050et_nat(some_set_nat,aa_nat_set_nat(Uu,Uua)) ).

% ATP.lambda_252
tff(fact_10458_ATP_Olambda__253,axiom,
    ! [Uu: fun_int_set_nat,Uua: int] : aa_int9046179520680703296et_nat(aTP_Lamm_xo(Uu),Uua) = aa_set6119337032432945050et_nat(some_set_nat,aa_int_set_nat(Uu,Uua)) ).

% ATP.lambda_253
tff(fact_10459_ATP_Olambda__254,axiom,
    ! [Uu: fun_VE3114949619472531401T_VEBT,Uua: vEBT_VEBT] : aa_VEB8154608084236005916T_VEBT(aTP_Lamm_acc(Uu),Uua) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_VEB5888937730619358530T_VEBT(Uu,Uua)) ).

% ATP.lambda_254
tff(fact_10460_ATP_Olambda__255,axiom,
    ! [Uu: fun_VE8613710830879766659t_real,Uua: vEBT_VEBT] : aa_VEB5302157898331830618t_real(aTP_Lamm_acg(Uu),Uua) = aa_lis3769085826154824284t_real(set_real2,aa_VEB6922606862176952244t_real(Uu,Uua)) ).

% ATP.lambda_255
tff(fact_10461_ATP_Olambda__256,axiom,
    ! [Uu: fun_VE3624423849788874412t_bool,Uua: vEBT_VEBT] : aa_VEB3235182236201257599t_bool(aTP_Lamm_ack(Uu),Uua) = aa_lis3587183946281582166t_bool(set_bool2,aa_VEB5603231326792150437t_bool(Uu,Uua)) ).

% ATP.lambda_256
tff(fact_10462_ATP_Olambda__257,axiom,
    ! [Uu: fun_re8294352169103142759T_VEBT,Uua: real] : aa_rea5591236203076396202T_VEBT(aTP_Lamm_ace(Uu),Uua) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_rea924616363743086800T_VEBT(Uu,Uua)) ).

% ATP.lambda_257
tff(fact_10463_ATP_Olambda__258,axiom,
    ! [Uu: fun_real_list_real,Uua: real] : aa_real_set_real(aTP_Lamm_aci(Uu),Uua) = aa_lis3769085826154824284t_real(set_real2,aa_real_list_real(Uu,Uua)) ).

% ATP.lambda_258
tff(fact_10464_ATP_Olambda__259,axiom,
    ! [Uu: fun_bo1245769998338026534T_VEBT,Uua: bool] : aa_boo8213446378722761977T_VEBT(aTP_Lamm_acf(Uu),Uua) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_boo8172854069374175647T_VEBT(Uu,Uua)) ).

% ATP.lambda_259
tff(fact_10465_ATP_Olambda__260,axiom,
    ! [Uu: fun_bool_list_real,Uua: bool] : aa_bool_set_real(aTP_Lamm_acj(Uu),Uua) = aa_lis3769085826154824284t_real(set_real2,aa_bool_list_real(Uu,Uua)) ).

% ATP.lambda_260
tff(fact_10466_ATP_Olambda__261,axiom,
    ! [Uu: fun_na8123536816256238275T_VEBT,Uua: nat] : aa_nat_set_VEBT_VEBT(aTP_Lamm_acd(Uu),Uua) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,aa_nat1721457043056498732T_VEBT(Uu,Uua)) ).

% ATP.lambda_261
tff(fact_10467_ATP_Olambda__262,axiom,
    ! [Uu: fun_nat_list_real,Uua: nat] : aa_nat_set_real(aTP_Lamm_ach(Uu),Uua) = aa_lis3769085826154824284t_real(set_real2,aa_nat_list_real(Uu,Uua)) ).

% ATP.lambda_262
tff(fact_10468_ATP_Olambda__263,axiom,
    ! [Uu: fun_nat_list_bool,Uua: nat] : aa_nat_set_bool(aTP_Lamm_acl(Uu),Uua) = aa_lis3587183946281582166t_bool(set_bool2,aa_nat_list_bool(Uu,Uua)) ).

% ATP.lambda_263
tff(fact_10469_ATP_Olambda__264,axiom,
    ! [Uu: fun_Co3766629010929971184T_VEBT,Uua: code_integer] : aa_Cod8307765772368743006T_VEBT(aTP_Lamm_vs(Uu),Uua) = insert_VEBT_VEBT(aa_Cod6507154402108480601T_VEBT(Uu,Uua)) ).

% ATP.lambda_264
tff(fact_10470_ATP_Olambda__265,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: code_integer] : aa_Cod3429118290910873308t_real(aTP_Lamm_vw(Uu),Uua) = insert_real(aa_Code_integer_real(Uu,Uua)) ).

% ATP.lambda_265
tff(fact_10471_ATP_Olambda__266,axiom,
    ! [Uu: fun_co4340422964966740447T_VEBT,Uua: complex] : aa_com5204220621766640175T_VEBT(aTP_Lamm_vr(Uu),Uua) = insert_VEBT_VEBT(aa_complex_VEBT_VEBT(Uu,Uua)) ).

% ATP.lambda_266
tff(fact_10472_ATP_Olambda__267,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_com4038042666021023277t_real(aTP_Lamm_vv(Uu),Uua) = insert_real(aa_complex_real(Uu,Uua)) ).

% ATP.lambda_267
tff(fact_10473_ATP_Olambda__268,axiom,
    ! [Uu: fun_nat_VEBT_VEBT,Uua: nat] : aa_nat5437107641380448849T_VEBT(aTP_Lamm_vp(Uu),Uua) = insert_VEBT_VEBT(aa_nat_VEBT_VEBT(Uu,Uua)) ).

% ATP.lambda_268
tff(fact_10474_ATP_Olambda__269,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat3311850034669319503t_real(aTP_Lamm_vt(Uu),Uua) = insert_real(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_269
tff(fact_10475_ATP_Olambda__270,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] : aa_nat3992033916769250705t_bool(aTP_Lamm_vx(Uu),Uua) = insert_bool(aa_nat_bool(Uu,Uua)) ).

% ATP.lambda_270
tff(fact_10476_ATP_Olambda__271,axiom,
    ! [Uu: fun_int_VEBT_VEBT,Uua: int] : aa_int5881555398370355885T_VEBT(aTP_Lamm_vq(Uu),Uua) = insert_VEBT_VEBT(aa_int_VEBT_VEBT(Uu,Uua)) ).

% ATP.lambda_271
tff(fact_10477_ATP_Olambda__272,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int1785798226927626155t_real(aTP_Lamm_vu(Uu),Uua) = insert_real(aa_int_real(Uu,Uua)) ).

% ATP.lambda_272
tff(fact_10478_ATP_Olambda__273,axiom,
    ! [Uu: fun_int_bool,Uua: int] : aa_int2397630360903011821t_bool(aTP_Lamm_vy(Uu),Uua) = insert_bool(aa_int_bool(Uu,Uua)) ).

% ATP.lambda_273
tff(fact_10479_ATP_Olambda__274,axiom,
    ! [Uu: fun_complex_bool,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_wc(Uu),Uua))
    <=> ~ pp(aa_complex_bool(Uu,Uua)) ) ).

% ATP.lambda_274
tff(fact_10480_ATP_Olambda__275,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_wd(Uu),Uua))
    <=> ~ pp(aa_list_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_10481_ATP_Olambda__276,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_we(Uu),Uua))
    <=> ~ pp(aa_set_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_10482_ATP_Olambda__277,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_wf(Uu),Uua))
    <=> ~ pp(aa_nat_bool(Uu,Uua)) ) ).

% ATP.lambda_277
tff(fact_10483_ATP_Olambda__278,axiom,
    ! [Uu: fun_int_bool,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_wg(Uu),Uua))
    <=> ~ pp(aa_int_bool(Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_10484_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_qn,Uu),Uua) = aa_nat_nat(suc,Uu) ).

% ATP.lambda_279
tff(fact_10485_ATP_Olambda__280,axiom,
    ! [Uu: nat,Uua: bool] : aa_bool_nat(aa_nat_fun_bool_nat(aTP_Lamm_qo,Uu),Uua) = aa_nat_nat(suc,Uu) ).

% ATP.lambda_280
tff(fact_10486_ATP_Olambda__281,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qp,Uu),Uua) = aa_nat_nat(suc,Uu) ).

% ATP.lambda_281
tff(fact_10487_ATP_Olambda__282,axiom,
    ! [Uu: nat,Uua: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_qq,Uu),Uua) = aa_nat_nat(suc,Uu) ).

% ATP.lambda_282
tff(fact_10488_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ajn(Uu),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_283
tff(fact_10489_ATP_Olambda__284,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ajc(Uu),Uua),Uub) = aa_real_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_284
tff(fact_10490_ATP_Olambda__285,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ajs(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uub),numera6620942414471956472nteger(Uu))),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_285
tff(fact_10491_ATP_Olambda__286,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ajq(Uu),Uua),Uub) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu)))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_286
tff(fact_10492_ATP_Olambda__287,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ajr(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),aa_int7462565939943104719nt_int(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_287
tff(fact_10493_ATP_Olambda__288,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_adu(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub)),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_288
tff(fact_10494_ATP_Olambda__289,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_adz(Uu),Uua),Uub) = aa_real_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uu),Uub),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_289
tff(fact_10495_ATP_Olambda__290,axiom,
    ! [Uu: code_integer,Uua: fun_Co1900716531650368264omplex,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ady(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uu),Uub)),aa_Cod5621350732091595593omplex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_290
tff(fact_10496_ATP_Olambda__291,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] : aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_aed(Uu),Uua),Uub) = aa_real_real(if_real(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uu),Uub),aa_Code_integer_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_291
tff(fact_10497_ATP_Olambda__292,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_aec(Uu),Uua),Uub) = aa_real_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_292
tff(fact_10498_ATP_Olambda__293,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_adv(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_293
tff(fact_10499_ATP_Olambda__294,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aea(Uu),Uua),Uub) = aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_294
tff(fact_10500_ATP_Olambda__295,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_adw(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub)),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_295
tff(fact_10501_ATP_Olambda__296,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_adx(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_296
tff(fact_10502_ATP_Olambda__297,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_aeb(Uu),Uua),Uub) = aa_real_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_297
tff(fact_10503_ATP_Olambda__298,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_adk(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu)),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_298
tff(fact_10504_ATP_Olambda__299,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_adp(Uu),Uua),Uub) = aa_real_real(if_real(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,Uub),Uu),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_299
tff(fact_10505_ATP_Olambda__300,axiom,
    ! [Uu: code_integer,Uua: fun_Co1900716531650368264omplex,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_fun7600406554973022798omplex(aTP_Lamm_ado(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),Uu)),aa_Cod5621350732091595593omplex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_300
tff(fact_10506_ATP_Olambda__301,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] : aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_adt(Uu),Uua),Uub) = aa_real_real(if_real(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),Uu),aa_Code_integer_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_301
tff(fact_10507_ATP_Olambda__302,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ads(Uu),Uua),Uub) = aa_real_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_302
tff(fact_10508_ATP_Olambda__303,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_adl(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_303
tff(fact_10509_ATP_Olambda__304,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adq(Uu),Uua),Uub) = aa_real_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_304
tff(fact_10510_ATP_Olambda__305,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_adm(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_305
tff(fact_10511_ATP_Olambda__306,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_adn(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_306
tff(fact_10512_ATP_Olambda__307,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_adr(Uu),Uua),Uub) = aa_real_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_307
tff(fact_10513_ATP_Olambda__308,axiom,
    ! [Uu: fun_VEBT_VEBT_bool,Uua: vEBT_VEBT,Uub: set_VEBT_VEBT] : aa_set3061394203342972348T_VEBT(aa_VEB8084284176465204155T_VEBT(aTP_Lamm_rl(Uu),Uua),Uub) = if_set_VEBT_VEBT(aa_VEBT_VEBT_bool(Uu,Uua),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Uua),Uub),Uub) ).

% ATP.lambda_308
tff(fact_10514_ATP_Olambda__309,axiom,
    ! [Uu: fun_Co8822979462182099027r_bool,Uua: code_integer,Uub: set_Code_integer] : aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(aTP_Lamm_rn(Uu),Uua),Uub) = if_set_Code_integer(aa_Code_integer_bool(Uu,Uua),insert_Code_integer(Uua,Uub),Uub) ).

% ATP.lambda_309
tff(fact_10515_ATP_Olambda__310,axiom,
    ! [Uu: fun_complex_bool,Uua: complex,Uub: set_complex] : aa_set8520398081763872006omplex(aa_com910761927370442929omplex(aTP_Lamm_rm(Uu),Uua),Uub) = if_set_complex(aa_complex_bool(Uu,Uua),insert_complex(Uua,Uub),Uub) ).

% ATP.lambda_310
tff(fact_10516_ATP_Olambda__311,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: set_real] : aa_set_real_set_real(aa_rea4059111765087356331t_real(aTP_Lamm_ro(Uu),Uua),Uub) = if_set_real(aa_real_bool(Uu,Uua),aa_set_real_set_real(insert_real(Uua),Uub),Uub) ).

% ATP.lambda_311
tff(fact_10517_ATP_Olambda__312,axiom,
    ! [Uu: fun_bool_bool,Uua: bool,Uub: set_bool] : aa_set_bool_set_bool(aa_boo5450306590482981214t_bool(aTP_Lamm_rp(Uu),Uua),Uub) = if_set_bool(aa_bool_bool(Uu,Uua),aa_set_bool_set_bool(insert_bool(Uua),Uub),Uub) ).

% ATP.lambda_312
tff(fact_10518_ATP_Olambda__313,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: set_nat] : aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(aTP_Lamm_rq(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(if_set_nat(aa_nat_bool(Uu,Uua)),aa_set_nat_set_nat(insert_nat(Uua),Uub)),Uub) ).

% ATP.lambda_313
tff(fact_10519_ATP_Olambda__314,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: set_int] : aa_set_int_set_int(aa_int7955932875426912299et_int(aTP_Lamm_rr(Uu),Uua),Uub) = aa_set_int_set_int(aa_set4234024024286713313et_int(if_set_int(aa_int_bool(Uu,Uua)),aa_set_int_set_int(insert_int(Uua),Uub)),Uub) ).

% ATP.lambda_314
tff(fact_10520_ATP_Olambda__315,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6792494735270171281omplex(aTP_Lamm_ahu(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_VEBT_VEBT_bool(Uua,Uub)),aa_VEBT_VEBT_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_315
tff(fact_10521_ATP_Olambda__316,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun2072160908118572431T_real(aTP_Lamm_aia(Uu),Uua),Uub) = aa_real_real(if_real(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_316
tff(fact_10522_ATP_Olambda__317,axiom,
    ! [Uu: fun_Co1900716531650368264omplex,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_fun5444226037728322975omplex(aTP_Lamm_ahz(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_Code_integer_bool(Uua,Uub)),aa_Cod5621350732091595593omplex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_317
tff(fact_10523_ATP_Olambda__318,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] : aa_Code_integer_real(aa_fun1813952354503624093r_real(aTP_Lamm_aie(Uu),Uua),Uub) = aa_real_real(if_real(aa_Code_integer_bool(Uua,Uub),aa_Code_integer_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_318
tff(fact_10524_ATP_Olambda__319,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_aid(Uu),Uua),Uub) = aa_real_real(if_real(aa_complex_bool(Uua,Uub),aa_complex_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_319
tff(fact_10525_ATP_Olambda__320,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_ahv(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_real_bool(Uua,Uub)),aa_real_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_320
tff(fact_10526_ATP_Olambda__321,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_aib(Uu),Uua),Uub) = aa_real_real(if_real(aa_real_bool(Uua,Uub),aa_real_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_321
tff(fact_10527_ATP_Olambda__322,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_ahw(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_nat_bool(Uua,Uub)),aa_nat_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_322
tff(fact_10528_ATP_Olambda__323,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_ahx(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(if_complex(aa_int_bool(Uua,Uub)),aa_int_complex(Uu,Uub)),zero_zero_complex) ).

% ATP.lambda_323
tff(fact_10529_ATP_Olambda__324,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_aic(Uu),Uua),Uub) = aa_real_real(if_real(aa_int_bool(Uua,Uub),aa_int_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_324
tff(fact_10530_ATP_Olambda__325,axiom,
    ! [Uu: fun_VE2006399705683203104omplex,Uua: complex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_com2862003112617944841omplex(aTP_Lamm_agz(Uu),Uua),Uub) = aa_complex_complex(aa_VEB5608662635949521omplex(Uu,Uub),Uua) ).

% ATP.lambda_325
tff(fact_10531_ATP_Olambda__326,axiom,
    ! [Uu: fun_VE597728521808591204at_nat,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_agc(Uu),Uua),Uub) = aa_nat_nat(aa_VEB1122092716774696213at_nat(Uu,Uub),Uua) ).

% ATP.lambda_326
tff(fact_10532_ATP_Olambda__327,axiom,
    ! [Uu: fun_Co8905303033605938227omplex,Uua: complex,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_com7552888802784664052omplex(aTP_Lamm_ahr(Uu),Uua),Uub) = aa_complex_complex(aa_Cod3397930797123873268omplex(Uu,Uub),Uua) ).

% ATP.lambda_327
tff(fact_10533_ATP_Olambda__328,axiom,
    ! [Uu: fun_Co3312552843764780407at_nat,Uua: nat,Uub: code_integer] : aa_Code_integer_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_agu(Uu),Uua),Uub) = aa_nat_nat(aa_Cod8670293052248272184at_nat(Uu,Uub),Uua) ).

% ATP.lambda_328
tff(fact_10534_ATP_Olambda__329,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_aeq(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_329
tff(fact_10535_ATP_Olambda__330,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_agp(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_330
tff(fact_10536_ATP_Olambda__331,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_ahd(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_331
tff(fact_10537_ATP_Olambda__332,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_agg(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_332
tff(fact_10538_ATP_Olambda__333,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ahh(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_333
tff(fact_10539_ATP_Olambda__334,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_aet(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_334
tff(fact_10540_ATP_Olambda__335,axiom,
    ! [Uu: fun_na6128946301132549280et_nat,Uua: nat,Uub: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_aao(Uu),Uua),Uub) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(Uu,Uub),Uua) ).

% ATP.lambda_335
tff(fact_10541_ATP_Olambda__336,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aen(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_336
tff(fact_10542_ATP_Olambda__337,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_ahm(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_337
tff(fact_10543_ATP_Olambda__338,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_agk(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_338
tff(fact_10544_ATP_Olambda__339,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aew(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_339
tff(fact_10545_ATP_Olambda__340,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_alg(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_alf(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_340
tff(fact_10546_ATP_Olambda__341,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ale(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ald(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_341
tff(fact_10547_ATP_Olambda__342,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_alc(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_alb(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_342
tff(fact_10548_ATP_Olambda__343,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ala(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_akz(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_343
tff(fact_10549_ATP_Olambda__344,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_aky(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_akx(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_344
tff(fact_10550_ATP_Olambda__345,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_akv(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_aku(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_345
tff(fact_10551_ATP_Olambda__346,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_aex(Uu),Uua),Uub) = aa_set_int_int(groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_aew(Uu),Uub)),Uua) ).

% ATP.lambda_346
tff(fact_10552_ATP_Olambda__347,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_aeu(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_aet(Uu),Uub)),Uua) ).

% ATP.lambda_347
tff(fact_10553_ATP_Olambda__348,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_aer(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_aeq(Uu),Uub),Uua) ).

% ATP.lambda_348
tff(fact_10554_ATP_Olambda__349,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_aeo(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aen(Uu),Uub)),Uua) ).

% ATP.lambda_349
tff(fact_10555_ATP_Olambda__350,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aim(Uu),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(power_power_real(Uu),Uub)) ).

% ATP.lambda_350
tff(fact_10556_ATP_Olambda__351,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_alr(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_351
tff(fact_10557_ATP_Olambda__352,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_als(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(groups4561878855575611511st_nat,Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_352
tff(fact_10558_ATP_Olambda__353,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_nq(Uu),Uua),Uub))
    <=> ( ( size_s6755466524823107622T_VEBT(Uub) = Uua )
        & distinct_VEBT_VEBT(Uub)
        & pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uub)),Uu)) ) ) ).

% ATP.lambda_353
tff(fact_10559_ATP_Olambda__354,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( pp(aa_lis9036377640098331692r_bool(aa_nat3680331675976215733r_bool(aTP_Lamm_ns(Uu),Uua),Uub))
    <=> ( ( size_s3445333598471063425nteger(Uub) = Uua )
        & distin1543349897113766820nteger(Uub)
        & pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu)) ) ) ).

% ATP.lambda_354
tff(fact_10560_ATP_Olambda__355,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_nr(Uu),Uua),Uub))
    <=> ( ( size_s3451745648224563538omplex(Uub) = Uua )
        & distinct_complex(Uub)
        & pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)) ) ) ).

% ATP.lambda_355
tff(fact_10561_ATP_Olambda__356,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( pp(aa_list_real_bool(aa_nat4570152432052498854l_bool(aTP_Lamm_nt(Uu),Uua),Uub))
    <=> ( ( aa_list_real_nat(size_size_list_real,Uub) = Uua )
        & pp(aa_list_real_bool(distinct_real,Uub))
        & pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Uub)),Uu)) ) ) ).

% ATP.lambda_356
tff(fact_10562_ATP_Olambda__357,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_nu(Uu),Uua),Uub))
    <=> ( ( aa_list_bool_nat(size_size_list_bool,Uub) = Uua )
        & pp(aa_list_bool_bool(distinct_bool,Uub))
        & pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Uub)),Uu)) ) ) ).

% ATP.lambda_357
tff(fact_10563_ATP_Olambda__358,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_nv(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua )
        & pp(aa_list_nat_bool(distinct_nat,Uub))
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Uub)),Uu)) ) ) ).

% ATP.lambda_358
tff(fact_10564_ATP_Olambda__359,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_nw(Uu),Uua),Uub))
    <=> ( ( aa_list_int_nat(size_size_list_int,Uub) = Uua )
        & pp(aa_list_int_bool(distinct_int,Uub))
        & pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Uub)),Uu)) ) ) ).

% ATP.lambda_359
tff(fact_10565_ATP_Olambda__360,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_lis561826058711847838T_bool(aTP_Lamm_fx(Uu),Uua),Uub))
    <=> ( ( size_s6755466524823107622T_VEBT(Uub) = Uu )
        & pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uub)),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uua))) ) ) ).

% ATP.lambda_360
tff(fact_10566_ATP_Olambda__361,axiom,
    ! [Uu: nat,Uua: list_real,Uub: list_real] :
      ( pp(aa_list_real_bool(aa_lis8290120702490534970l_bool(aTP_Lamm_fy(Uu),Uua),Uub))
    <=> ( ( aa_list_real_nat(size_size_list_real,Uub) = Uu )
        & pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Uub)),aa_lis3769085826154824284t_real(set_real2,Uua))) ) ) ).

% ATP.lambda_361
tff(fact_10567_ATP_Olambda__362,axiom,
    ! [Uu: nat,Uua: list_bool,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_lis3148754934519358302l_bool(aTP_Lamm_fz(Uu),Uua),Uub))
    <=> ( ( aa_list_bool_nat(size_size_list_bool,Uub) = Uu )
        & pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Uub)),aa_lis3587183946281582166t_bool(set_bool2,Uua))) ) ) ).

% ATP.lambda_362
tff(fact_10568_ATP_Olambda__363,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_ga(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Uub)),aa_list_nat_set_nat(set_nat2,Uua))) ) ) ).

% ATP.lambda_363
tff(fact_10569_ATP_Olambda__364,axiom,
    ! [Uu: nat,Uua: list_int,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(aTP_Lamm_gb(Uu),Uua),Uub))
    <=> ( ( aa_list_int_nat(size_size_list_int,Uub) = Uu )
        & pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Uub)),aa_list_int_set_int(set_int2,Uua))) ) ) ).

% ATP.lambda_364
tff(fact_10570_ATP_Olambda__365,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_fj(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua)) ) ) ).

% ATP.lambda_365
tff(fact_10571_ATP_Olambda__366,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( pp(aa_lis9036377640098331692r_bool(aa_nat3680331675976215733r_bool(aTP_Lamm_fl(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3445333598471063425nteger(Uub)),Uua)) ) ) ).

% ATP.lambda_366
tff(fact_10572_ATP_Olambda__367,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_fk(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua)) ) ) ).

% ATP.lambda_367
tff(fact_10573_ATP_Olambda__368,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( pp(aa_list_real_bool(aa_nat4570152432052498854l_bool(aTP_Lamm_fm(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_real_nat(size_size_list_real,Uub)),Uua)) ) ) ).

% ATP.lambda_368
tff(fact_10574_ATP_Olambda__369,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fn(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_bool_nat(size_size_list_bool,Uub)),Uua)) ) ) ).

% ATP.lambda_369
tff(fact_10575_ATP_Olambda__370,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_fo(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Uub)),Uua)) ) ) ).

% ATP.lambda_370
tff(fact_10576_ATP_Olambda__371,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_fp(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_list_int_nat(size_size_list_int,Uub)),Uua)) ) ) ).

% ATP.lambda_371
tff(fact_10577_ATP_Olambda__372,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_fq(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uub)),Uu))
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_372
tff(fact_10578_ATP_Olambda__373,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( pp(aa_lis9036377640098331692r_bool(aa_nat3680331675976215733r_bool(aTP_Lamm_fs(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu))
        & ( size_s3445333598471063425nteger(Uub) = Uua ) ) ) ).

% ATP.lambda_373
tff(fact_10579_ATP_Olambda__374,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_fr(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_374
tff(fact_10580_ATP_Olambda__375,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( pp(aa_list_real_bool(aa_nat4570152432052498854l_bool(aTP_Lamm_ft(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Uub)),Uu))
        & ( aa_list_real_nat(size_size_list_real,Uub) = Uua ) ) ) ).

% ATP.lambda_375
tff(fact_10581_ATP_Olambda__376,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_fu(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(ord_less_eq_set_bool(aa_lis3587183946281582166t_bool(set_bool2,Uub)),Uu))
        & ( aa_list_bool_nat(size_size_list_bool,Uub) = Uua ) ) ) ).

% ATP.lambda_376
tff(fact_10582_ATP_Olambda__377,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_fv(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Uub)),Uu))
        & ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_377
tff(fact_10583_ATP_Olambda__378,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_fw(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Uub)),Uu))
        & ( aa_list_int_nat(size_size_list_int,Uub) = Uua ) ) ) ).

% ATP.lambda_378
tff(fact_10584_ATP_Olambda__379,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_alq(Uu),Uua),Uub))
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_379
tff(fact_10585_ATP_Olambda__380,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aln(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_380
tff(fact_10586_ATP_Olambda__381,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_alm(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),pow_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),Uu))))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_381
tff(fact_10587_ATP_Olambda__382,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_alo(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_382
tff(fact_10588_ATP_Olambda__383,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).

% ATP.lambda_383
tff(fact_10589_ATP_Olambda__384,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_rw(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        | pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua)) ) ) ).

% ATP.lambda_384
tff(fact_10590_ATP_Olambda__385,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_ry(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        | pp(aa_set_complex_bool(member_complex(Uub),Uua)) ) ) ).

% ATP.lambda_385
tff(fact_10591_ATP_Olambda__386,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_rx(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        | pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_386
tff(fact_10592_ATP_Olambda__387,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_rz(Uu),Uua),Uub))
    <=> ( pp(aa_set_list_nat_bool(member_list_nat(Uub),Uu))
        | pp(aa_set_list_nat_bool(member_list_nat(Uub),Uua)) ) ) ).

% ATP.lambda_387
tff(fact_10593_ATP_Olambda__388,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_sa(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        | pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_388
tff(fact_10594_ATP_Olambda__389,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_sc(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        | pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_389
tff(fact_10595_ATP_Olambda__390,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_sb(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        | pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_390
tff(fact_10596_ATP_Olambda__391,axiom,
    ! [Uu: vEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_kc(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua)) ) ) ).

% ATP.lambda_391
tff(fact_10597_ATP_Olambda__392,axiom,
    ! [Uu: complex,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_ke(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_complex_bool(member_complex(Uub),Uua)) ) ) ).

% ATP.lambda_392
tff(fact_10598_ATP_Olambda__393,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_kd(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_393
tff(fact_10599_ATP_Olambda__394,axiom,
    ! [Uu: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_kf(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_list_nat_bool(member_list_nat(Uub),Uua)) ) ) ).

% ATP.lambda_394
tff(fact_10600_ATP_Olambda__395,axiom,
    ! [Uu: bool,Uua: set_bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_set6670506438128111678l_bool(aTP_Lamm_kb(Uu),Uua),Uub))
    <=> ( ( pp(Uub)
        <=> pp(Uu) )
        | pp(aa_set_bool_bool(member_bool(Uub),Uua)) ) ) ).

% ATP.lambda_395
tff(fact_10601_ATP_Olambda__396,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_kg(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_396
tff(fact_10602_ATP_Olambda__397,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_kh(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_397
tff(fact_10603_ATP_Olambda__398,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_ki(Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_398
tff(fact_10604_ATP_Olambda__399,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ajw(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_399
tff(fact_10605_ATP_Olambda__400,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_rs(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_400
tff(fact_10606_ATP_Olambda__401,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ajx(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_401
tff(fact_10607_ATP_Olambda__402,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ru(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_402
tff(fact_10608_ATP_Olambda__403,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_rv(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_403
tff(fact_10609_ATP_Olambda__404,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_rt(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_404
tff(fact_10610_ATP_Olambda__405,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aw(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uub)) ) ) ).

% ATP.lambda_405
tff(fact_10611_ATP_Olambda__406,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bf(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_406
tff(fact_10612_ATP_Olambda__407,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer,Uub: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(aa_set8732061614394825996r_bool(aTP_Lamm_is(Uu),Uua),Uub))
    <=> ( minus_2355218937544613996nteger(Uub,Uu) = minus_2355218937544613996nteger(Uua,Uu) ) ) ).

% ATP.lambda_407
tff(fact_10613_ATP_Olambda__408,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: set_complex] :
      ( pp(aa_set_complex_bool(aa_set6592632205191297002x_bool(aTP_Lamm_ir(Uu),Uua),Uub))
    <=> ( minus_811609699411566653omplex(Uub,Uu) = minus_811609699411566653omplex(Uua,Uu) ) ) ).

% ATP.lambda_408
tff(fact_10614_ATP_Olambda__409,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_it(Uu),Uua),Uub))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Uub),Uu) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Uua),Uu) ) ) ).

% ATP.lambda_409
tff(fact_10615_ATP_Olambda__410,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: set_int] :
      ( pp(aa_set_int_bool(aa_set7694327796961020398t_bool(aTP_Lamm_iq(Uu),Uua),Uub))
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,Uub),Uu) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,Uua),Uu) ) ) ).

% ATP.lambda_410
tff(fact_10616_ATP_Olambda__411,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aTP_Lamm_bg(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & pp(aa_VEBT_VEBT_bool(Uua,Uub)) ) ) ).

% ATP.lambda_411
tff(fact_10617_ATP_Olambda__412,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8875953811088571772r_bool(aTP_Lamm_ahy(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uub),Uu))
        & pp(aa_Code_integer_bool(Uua,Uub)) ) ) ).

% ATP.lambda_412
tff(fact_10618_ATP_Olambda__413,axiom,
    ! [Uu: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_bi(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_413
tff(fact_10619_ATP_Olambda__414,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_bh(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_414
tff(fact_10620_ATP_Olambda__415,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_bj(Uu),Uua),Uub))
    <=> ( pp(aa_set_list_nat_bool(member_list_nat(Uub),Uu))
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_415
tff(fact_10621_ATP_Olambda__416,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_bk(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_416
tff(fact_10622_ATP_Olambda__417,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_bl(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_417
tff(fact_10623_ATP_Olambda__418,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_bm(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_418
tff(fact_10624_ATP_Olambda__419,axiom,
    ! [Uu: fun_VEBT_VEBT_bool,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_sd(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua))
        & pp(aa_VEBT_VEBT_bool(Uu,Uub)) ) ) ).

% ATP.lambda_419
tff(fact_10625_ATP_Olambda__420,axiom,
    ! [Uu: fun_complex_bool,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_sf(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uua))
        & pp(aa_complex_bool(Uu,Uub)) ) ) ).

% ATP.lambda_420
tff(fact_10626_ATP_Olambda__421,axiom,
    ! [Uu: fun_real_bool,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_se(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uua))
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_421
tff(fact_10627_ATP_Olambda__422,axiom,
    ! [Uu: fun_list_nat_bool,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_sg(Uu),Uua),Uub))
    <=> ( pp(aa_set_list_nat_bool(member_list_nat(Uub),Uua))
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_422
tff(fact_10628_ATP_Olambda__423,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_sh(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua))
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_423
tff(fact_10629_ATP_Olambda__424,axiom,
    ! [Uu: fun_nat_bool,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_si(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uua))
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_424
tff(fact_10630_ATP_Olambda__425,axiom,
    ! [Uu: fun_int_bool,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_sj(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uua))
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_425
tff(fact_10631_ATP_Olambda__426,axiom,
    ! [Uu: fun_VEBT_VEBT_bool,Uua: vEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aTP_Lamm_kr(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_VEBT_VEBT_bool(Uu,Uub)) ) ) ).

% ATP.lambda_426
tff(fact_10632_ATP_Olambda__427,axiom,
    ! [Uu: fun_complex_bool,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_ks(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_complex_bool(Uu,Uub)) ) ) ).

% ATP.lambda_427
tff(fact_10633_ATP_Olambda__428,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_kv(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_428
tff(fact_10634_ATP_Olambda__429,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_kt(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_429
tff(fact_10635_ATP_Olambda__430,axiom,
    ! [Uu: fun_bool_bool,Uua: bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_kw(Uu),Uua),Uub))
    <=> ( ( pp(Uua)
        <=> pp(Uub) )
        & pp(aa_bool_bool(Uu,Uub)) ) ) ).

% ATP.lambda_430
tff(fact_10636_ATP_Olambda__431,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_ku(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_431
tff(fact_10637_ATP_Olambda__432,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_kx(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_432
tff(fact_10638_ATP_Olambda__433,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ky(Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_433
tff(fact_10639_ATP_Olambda__434,axiom,
    ! [Uu: fun_VEBT_VEBT_bool,Uua: vEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aTP_Lamm_kj(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_VEBT_VEBT_bool(Uu,Uub)) ) ) ).

% ATP.lambda_434
tff(fact_10640_ATP_Olambda__435,axiom,
    ! [Uu: fun_complex_bool,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_kk(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_complex_bool(Uu,Uub)) ) ) ).

% ATP.lambda_435
tff(fact_10641_ATP_Olambda__436,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_kn(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_real_bool(Uu,Uub)) ) ) ).

% ATP.lambda_436
tff(fact_10642_ATP_Olambda__437,axiom,
    ! [Uu: fun_list_nat_bool,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_kl(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_list_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_437
tff(fact_10643_ATP_Olambda__438,axiom,
    ! [Uu: fun_bool_bool,Uua: bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_ko(Uu),Uua),Uub))
    <=> ( ( pp(Uub)
        <=> pp(Uua) )
        & pp(aa_bool_bool(Uu,Uub)) ) ) ).

% ATP.lambda_438
tff(fact_10644_ATP_Olambda__439,axiom,
    ! [Uu: fun_set_nat_bool,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_km(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_set_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_439
tff(fact_10645_ATP_Olambda__440,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_kp(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_nat_bool(Uu,Uub)) ) ) ).

% ATP.lambda_440
tff(fact_10646_ATP_Olambda__441,axiom,
    ! [Uu: fun_int_bool,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_kq(Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa_int_bool(Uu,Uub)) ) ) ).

% ATP.lambda_441
tff(fact_10647_ATP_Olambda__442,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aTP_Lamm_ep(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ( aa_VEBT_VEBT_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_442
tff(fact_10648_ATP_Olambda__443,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_fb(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ( aa_VEBT_VEBT_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_443
tff(fact_10649_ATP_Olambda__444,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1900716531650368264omplex,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun5910925830913328781r_bool(aTP_Lamm_ez(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uub),Uu))
        & ( aa_Cod5621350732091595593omplex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_444
tff(fact_10650_ATP_Olambda__445,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun7028086442024493327r_bool(aTP_Lamm_lg(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uub),Uu))
        & ( aa_Code_integer_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_445
tff(fact_10651_ATP_Olambda__446,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_ex(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_446
tff(fact_10652_ATP_Olambda__447,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_lf(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_447
tff(fact_10653_ATP_Olambda__448,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_er(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_448
tff(fact_10654_ATP_Olambda__449,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_fd(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_449
tff(fact_10655_ATP_Olambda__450,axiom,
    ! [Uu: set_bool,Uua: fun_bool_complex,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun4000492451996535199l_bool(aTP_Lamm_le(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & ( aa_bool_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_450
tff(fact_10656_ATP_Olambda__451,axiom,
    ! [Uu: set_bool,Uua: fun_bool_real,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_lh(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & ( aa_bool_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_451
tff(fact_10657_ATP_Olambda__452,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_et(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_452
tff(fact_10658_ATP_Olambda__453,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_ff(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_453
tff(fact_10659_ATP_Olambda__454,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_ev(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_454
tff(fact_10660_ATP_Olambda__455,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_fh(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_455
tff(fact_10661_ATP_Olambda__456,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_du(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ( aa_VEBT_VEBT_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_456
tff(fact_10662_ATP_Olambda__457,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun1017860439744518025T_bool(aTP_Lamm_eg(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ( aa_VEBT_VEBT_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_457
tff(fact_10663_ATP_Olambda__458,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun7028086442024493327r_bool(aTP_Lamm_ee(Uu),Uua),Uub))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uub),Uu))
        & ( aa_Code_integer_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_458
tff(fact_10664_ATP_Olambda__459,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_ec(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_459
tff(fact_10665_ATP_Olambda__460,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_dw(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_460
tff(fact_10666_ATP_Olambda__461,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_ei(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_461
tff(fact_10667_ATP_Olambda__462,axiom,
    ! [Uu: set_bool,Uua: fun_bool_real,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun2512541629346179229l_bool(aTP_Lamm_li(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & ( aa_bool_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_462
tff(fact_10668_ATP_Olambda__463,axiom,
    ! [Uu: set_bool,Uua: fun_bool_rat,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun4510340986217973961l_bool(aTP_Lamm_lj(Uu),Uua),Uub))
    <=> ( pp(aa_set_bool_bool(member_bool(Uub),Uu))
        & ( aa_bool_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_463
tff(fact_10669_ATP_Olambda__464,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_dy(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_464
tff(fact_10670_ATP_Olambda__465,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_ek(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_465
tff(fact_10671_ATP_Olambda__466,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_ea(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_466
tff(fact_10672_ATP_Olambda__467,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( pp(aa_int_bool(aa_fun258058769140594373t_bool(aTP_Lamm_em(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_467
tff(fact_10673_ATP_Olambda__468,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_lk(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua))
        & ( aa_VEBT_VEBT_complex(Uu,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_468
tff(fact_10674_ATP_Olambda__469,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_lp(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua))
        & ( aa_VEBT_VEBT_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_469
tff(fact_10675_ATP_Olambda__470,axiom,
    ! [Uu: fun_complex_complex,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_lm(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uua))
        & ( aa_complex_complex(Uu,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_470
tff(fact_10676_ATP_Olambda__471,axiom,
    ! [Uu: fun_complex_real,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_lr(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uua))
        & ( aa_complex_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_471
tff(fact_10677_ATP_Olambda__472,axiom,
    ! [Uu: fun_real_complex,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ll(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uua))
        & ( aa_real_complex(Uu,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_472
tff(fact_10678_ATP_Olambda__473,axiom,
    ! [Uu: fun_real_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_lq(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uua))
        & ( aa_real_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_473
tff(fact_10679_ATP_Olambda__474,axiom,
    ! [Uu: fun_nat_complex,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ln(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uua))
        & ( aa_nat_complex(Uu,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_474
tff(fact_10680_ATP_Olambda__475,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_ls(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uua))
        & ( aa_nat_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_475
tff(fact_10681_ATP_Olambda__476,axiom,
    ! [Uu: fun_int_complex,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_lo(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uua))
        & ( aa_int_complex(Uu,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_476
tff(fact_10682_ATP_Olambda__477,axiom,
    ! [Uu: fun_int_real,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_lt(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uua))
        & ( aa_int_real(Uu,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_477
tff(fact_10683_ATP_Olambda__478,axiom,
    ! [Uu: fun_VEBT_VEBT_assn,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_lu(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua))
        & ( aa_VEBT_VEBT_assn(Uu,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_478
tff(fact_10684_ATP_Olambda__479,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_lz(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua))
        & ( aa_VEBT_VEBT_real(Uu,Uub) != one_one_real ) ) ) ).

% ATP.lambda_479
tff(fact_10685_ATP_Olambda__480,axiom,
    ! [Uu: fun_complex_assn,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_lw(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uua))
        & ( aa_complex_assn(Uu,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_480
tff(fact_10686_ATP_Olambda__481,axiom,
    ! [Uu: fun_complex_real,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_mb(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uua))
        & ( aa_complex_real(Uu,Uub) != one_one_real ) ) ) ).

% ATP.lambda_481
tff(fact_10687_ATP_Olambda__482,axiom,
    ! [Uu: fun_real_assn,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_lv(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uua))
        & ( aa_real_assn(Uu,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_482
tff(fact_10688_ATP_Olambda__483,axiom,
    ! [Uu: fun_real_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_ma(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uua))
        & ( aa_real_real(Uu,Uub) != one_one_real ) ) ) ).

% ATP.lambda_483
tff(fact_10689_ATP_Olambda__484,axiom,
    ! [Uu: fun_nat_assn,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_lx(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uua))
        & ( aa_nat_assn(Uu,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_484
tff(fact_10690_ATP_Olambda__485,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_mc(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uua))
        & ( aa_nat_real(Uu,Uub) != one_one_real ) ) ) ).

% ATP.lambda_485
tff(fact_10691_ATP_Olambda__486,axiom,
    ! [Uu: fun_int_assn,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_ly(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uua))
        & ( aa_int_assn(Uu,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_486
tff(fact_10692_ATP_Olambda__487,axiom,
    ! [Uu: fun_int_real,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_md(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uua))
        & ( aa_int_real(Uu,Uub) != one_one_real ) ) ) ).

% ATP.lambda_487
tff(fact_10693_ATP_Olambda__488,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: bool] : aa_bool_assn(aa_VEB1972364096475303967l_assn(aTP_Lamm_eo(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,Uub),aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Uu)))) ).

% ATP.lambda_488
tff(fact_10694_ATP_Olambda__489,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_be(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uub),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,Uu)))) ).

% ATP.lambda_489
tff(fact_10695_ATP_Olambda__490,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_av(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uub),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,Uu)))) ).

% ATP.lambda_490
tff(fact_10696_ATP_Olambda__491,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: list_VEBT_VEBTi,Uub: nat] : aa_nat_assn(aa_lis7348694692495225848t_assn(aTP_Lamm_pq(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(Uu,Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),size_s7982070591426661849_VEBTi(Uua)))) ).

% ATP.lambda_491
tff(fact_10697_ATP_Olambda__492,axiom,
    ! [Uu: array_bool,Uua: list_bool,Uub: nat] : aa_nat_assn(aa_lis1286645363786127574t_assn(aTP_Lamm_pn(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_bool(Uu,Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),aa_list_bool_nat(size_size_list_bool,Uua)))) ).

% ATP.lambda_492
tff(fact_10698_ATP_Olambda__493,axiom,
    ! [Uu: array_nat,Uua: list_nat,Uub: nat] : aa_nat_assn(aa_lis6041139530851742867t_assn(aTP_Lamm_po(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_nat(Uu,Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),aa_list_nat_nat(size_size_list_nat,Uua)))) ).

% ATP.lambda_493
tff(fact_10699_ATP_Olambda__494,axiom,
    ! [Uu: array_int,Uua: list_int,Uub: nat] : aa_nat_assn(aa_lis2816288184639765743t_assn(aTP_Lamm_pp(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_int(Uu,Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),aa_list_int_nat(size_size_list_int,Uua)))) ).

% ATP.lambda_494
tff(fact_10700_ATP_Olambda__495,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_lis3169797078229555565i_assn(aTP_Lamm_oy(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(Uu,Uua)),aa_bool_assn(pure_assn,fequal1389750275112723248_VEBTi(Uub,Uua))) ).

% ATP.lambda_495
tff(fact_10701_ATP_Olambda__496,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_set1728193350978918526T_bool(aTP_Lamm_jm(Uu),Uua),Uub))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uu))
        & ~ pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uub),Uua)) ) ) ).

% ATP.lambda_496
tff(fact_10702_ATP_Olambda__497,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_jo(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ~ pp(aa_set_complex_bool(member_complex(Uub),Uua)) ) ) ).

% ATP.lambda_497
tff(fact_10703_ATP_Olambda__498,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_jn(Uu),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
        & ~ pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).

% ATP.lambda_498
tff(fact_10704_ATP_Olambda__499,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_jp(Uu),Uua),Uub))
    <=> ( pp(aa_set_list_nat_bool(member_list_nat(Uub),Uu))
        & ~ pp(aa_set_list_nat_bool(member_list_nat(Uub),Uua)) ) ) ).

% ATP.lambda_499
tff(fact_10705_ATP_Olambda__500,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set9121396911425987952t_bool(aTP_Lamm_jq(Uu),Uua),Uub))
    <=> ( pp(aa_set_set_nat_bool(member_set_nat(Uub),Uu))
        & ~ pp(aa_set_set_nat_bool(member_set_nat(Uub),Uua)) ) ) ).

% ATP.lambda_500
tff(fact_10706_ATP_Olambda__501,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_js(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ~ pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).

% ATP.lambda_501
tff(fact_10707_ATP_Olambda__502,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_jr(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ~ pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).

% ATP.lambda_502
tff(fact_10708_ATP_Olambda__503,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aii(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu)),Uua)) ) ).

% ATP.lambda_503
tff(fact_10709_ATP_Olambda__504,axiom,
    ! [Uu: set_Pr7556676689462069481BT_nat,Uua: vEBT_VEBT,Uub: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(aa_set2648297115105225046t_bool(aTP_Lamm_dq,Uu),Uua),Uub))
    <=> member373505688050248522BT_nat(aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Uua),Uub),Uu) ) ).

% ATP.lambda_504
tff(fact_10710_ATP_Olambda__505,axiom,
    ! [Uu: set_Pr4811707699266497531nteger,Uua: code_integer,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(aa_set8960070006189174672r_bool(aTP_Lamm_ds,Uu),Uua),Uub))
    <=> member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Uua),Uub),Uu) ) ).

% ATP.lambda_505
tff(fact_10711_ATP_Olambda__506,axiom,
    ! [Uu: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
      ( pp(aa_num_bool(aa_nat_fun_num_bool(aa_set4109929515018738736m_bool(aTP_Lamm_dt,Uu),Uua),Uub))
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(Uua),Uub),Uu) ) ).

% ATP.lambda_506
tff(fact_10712_ATP_Olambda__507,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_set1256819327649670340t_bool(aTP_Lamm_dp,Uu),Uua),Uub))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub),Uu) ) ).

% ATP.lambda_507
tff(fact_10713_ATP_Olambda__508,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_set8965632145344517972t_bool(aTP_Lamm_dr,Uu),Uua),Uub))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Uua),Uub),Uu) ) ).

% ATP.lambda_508
tff(fact_10714_ATP_Olambda__509,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_acq(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_509
tff(fact_10715_ATP_Olambda__510,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_alp(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_510
tff(fact_10716_ATP_Olambda__511,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ajj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_511
tff(fact_10717_ATP_Olambda__512,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aif(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_512
tff(fact_10718_ATP_Olambda__513,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_di(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_513
tff(fact_10719_ATP_Olambda__514,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: vEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aTP_Lamm_gq(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(Uu,Uua)),aa_VEBT_VEBT_real(Uu,Uub))) ) ).

% ATP.lambda_514
tff(fact_10720_ATP_Olambda__515,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: vEBT_VEBT,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aTP_Lamm_gw(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(Uu,Uua)),aa_VEBT_VEBT_nat(Uu,Uub))) ) ).

% ATP.lambda_515
tff(fact_10721_ATP_Olambda__516,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gr(Uu),Uua),Uub))
    <=> pp(aa_bool_bool(aa_boo8970175072276269406l_bool(ord_less_eq_bool,aa_nat_bool(Uu,Uua)),aa_nat_bool(Uu,Uub))) ) ).

% ATP.lambda_516
tff(fact_10722_ATP_Olambda__517,axiom,
    ! [Uu: fun_nat_rat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gs(Uu),Uua),Uub))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,Uub))) ) ).

% ATP.lambda_517
tff(fact_10723_ATP_Olambda__518,axiom,
    ! [Uu: fun_nat_num,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gu(Uu),Uua),Uub))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(Uu,Uua)),aa_nat_num(Uu,Uub))) ) ).

% ATP.lambda_518
tff(fact_10724_ATP_Olambda__519,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gx(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uua)),aa_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_519
tff(fact_10725_ATP_Olambda__520,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gz(Uu),Uua),Uub))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,Uub))) ) ).

% ATP.lambda_520
tff(fact_10726_ATP_Olambda__521,axiom,
    ! [Uu: fun_int_rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_gt(Uu),Uua),Uub))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(Uu,Uua)),aa_int_rat(Uu,Uub))) ) ).

% ATP.lambda_521
tff(fact_10727_ATP_Olambda__522,axiom,
    ! [Uu: fun_int_num,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_gv(Uu),Uua),Uub))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(Uu,Uua)),aa_int_num(Uu,Uub))) ) ).

% ATP.lambda_522
tff(fact_10728_ATP_Olambda__523,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_gy(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(Uu,Uua)),aa_int_nat(Uu,Uub))) ) ).

% ATP.lambda_523
tff(fact_10729_ATP_Olambda__524,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_ng(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(Uu,Uub)),aa_VEBT_VEBT_real(Uua,Uub)) ).

% ATP.lambda_524
tff(fact_10730_ATP_Olambda__525,axiom,
    ! [Uu: fun_VEBT_VEBT_rat,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_ni(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_VEBT_VEBT_rat(Uu,Uub)),aa_VEBT_VEBT_rat(Uua,Uub)) ).

% ATP.lambda_525
tff(fact_10731_ATP_Olambda__526,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] : aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_mr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(Uu,Uub)),aa_Code_integer_real(Uua,Uub)) ).

% ATP.lambda_526
tff(fact_10732_ATP_Olambda__527,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: fun_Code_integer_rat,Uub: code_integer] : aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_mv(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Code_integer_rat(Uu,Uub)),aa_Code_integer_rat(Uua,Uub)) ).

% ATP.lambda_527
tff(fact_10733_ATP_Olambda__528,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_mq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uu,Uub)),aa_complex_real(Uua,Uub)) ).

% ATP.lambda_528
tff(fact_10734_ATP_Olambda__529,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_mu(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(Uu,Uub)),aa_complex_rat(Uua,Uub)) ).

% ATP.lambda_529
tff(fact_10735_ATP_Olambda__530,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_530
tff(fact_10736_ATP_Olambda__531,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nj(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uu,Uub)),aa_real_rat(Uua,Uub)) ).

% ATP.lambda_531
tff(fact_10737_ATP_Olambda__532,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_532
tff(fact_10738_ATP_Olambda__533,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ms(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_533
tff(fact_10739_ATP_Olambda__534,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_mw(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_534
tff(fact_10740_ATP_Olambda__535,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_mp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uu,Uub)),aa_int_real(Uua,Uub)) ).

% ATP.lambda_535
tff(fact_10741_ATP_Olambda__536,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_mt(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uu,Uub)),aa_int_rat(Uua,Uub)) ).

% ATP.lambda_536
tff(fact_10742_ATP_Olambda__537,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_mx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_537
tff(fact_10743_ATP_Olambda__538,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_afu(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_538
tff(fact_10744_ATP_Olambda__539,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_afv(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_539
tff(fact_10745_ATP_Olambda__540,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_afw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_540
tff(fact_10746_ATP_Olambda__541,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aar(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(Uu,Uub)),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_541
tff(fact_10747_ATP_Olambda__542,axiom,
    ! [Uu: fun_VE5569070535936029429omplex,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_my(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_VEBT_VEBT_complex(Uu,Uub)),aa_VEBT_VEBT_complex(Uua,Uub)) ).

% ATP.lambda_542
tff(fact_10748_ATP_Olambda__543,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_ne(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uu,Uub)),aa_VEBT_VEBT_real(Uua,Uub)) ).

% ATP.lambda_543
tff(fact_10749_ATP_Olambda__544,axiom,
    ! [Uu: fun_Co1900716531650368264omplex,Uua: fun_Co1900716531650368264omplex,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_fun7600406554973022798omplex(aTP_Lamm_nd(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_Cod5621350732091595593omplex(Uu,Uub)),aa_Cod5621350732091595593omplex(Uua,Uub)) ).

% ATP.lambda_544
tff(fact_10750_ATP_Olambda__545,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] : aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_mh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(Uu,Uub)),aa_Code_integer_real(Uua,Uub)) ).

% ATP.lambda_545
tff(fact_10751_ATP_Olambda__546,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: fun_Code_integer_rat,Uub: code_integer] : aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_ml(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(Uu,Uub)),aa_Code_integer_rat(Uua,Uub)) ).

% ATP.lambda_546
tff(fact_10752_ATP_Olambda__547,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_nc(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_547
tff(fact_10753_ATP_Olambda__548,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_mg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uu,Uub)),aa_complex_real(Uua,Uub)) ).

% ATP.lambda_548
tff(fact_10754_ATP_Olambda__549,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_mk(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(Uu,Uub)),aa_complex_rat(Uua,Uub)) ).

% ATP.lambda_549
tff(fact_10755_ATP_Olambda__550,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_mz(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(Uu,Uub)),aa_real_complex(Uua,Uub)) ).

% ATP.lambda_550
tff(fact_10756_ATP_Olambda__551,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_551
tff(fact_10757_ATP_Olambda__552,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_na(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_552
tff(fact_10758_ATP_Olambda__553,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_me(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_553
tff(fact_10759_ATP_Olambda__554,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mi(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_554
tff(fact_10760_ATP_Olambda__555,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_mm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_555
tff(fact_10761_ATP_Olambda__556,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_nb(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_int_complex(Uu,Uub)),aa_int_complex(Uua,Uub)) ).

% ATP.lambda_556
tff(fact_10762_ATP_Olambda__557,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_mf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uu,Uub)),aa_int_real(Uua,Uub)) ).

% ATP.lambda_557
tff(fact_10763_ATP_Olambda__558,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_mj(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Uu,Uub)),aa_int_rat(Uua,Uub)) ).

% ATP.lambda_558
tff(fact_10764_ATP_Olambda__559,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_mn(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_559
tff(fact_10765_ATP_Olambda__560,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_afd(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_560
tff(fact_10766_ATP_Olambda__561,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_wo(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
       => pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_561
tff(fact_10767_ATP_Olambda__562,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_wp(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
       => pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_562
tff(fact_10768_ATP_Olambda__563,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_wq(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
       => pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_563
tff(fact_10769_ATP_Olambda__564,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ws(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
       => pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_564
tff(fact_10770_ATP_Olambda__565,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_wr(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
       => pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_565
tff(fact_10771_ATP_Olambda__566,axiom,
    ! [Uu: fun_Co8822979462182099027r_bool,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8875953811088571772r_bool(aTP_Lamm_cw(Uu),Uua),Uub))
    <=> ( pp(aa_Code_integer_bool(Uu,Uub))
        | pp(aa_Code_integer_bool(Uua,Uub)) ) ) ).

% ATP.lambda_566
tff(fact_10772_ATP_Olambda__567,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_cv(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        | pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_567
tff(fact_10773_ATP_Olambda__568,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_cr(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        | pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_568
tff(fact_10774_ATP_Olambda__569,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_cs(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        | pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_569
tff(fact_10775_ATP_Olambda__570,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_ct(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        | pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_570
tff(fact_10776_ATP_Olambda__571,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_cu(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        | pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_571
tff(fact_10777_ATP_Olambda__572,axiom,
    ! [Uu: fun_Co8822979462182099027r_bool,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8875953811088571772r_bool(aTP_Lamm_cq(Uu),Uua),Uub))
    <=> ( pp(aa_Code_integer_bool(Uu,Uub))
        & pp(aa_Code_integer_bool(Uua,Uub)) ) ) ).

% ATP.lambda_572
tff(fact_10778_ATP_Olambda__573,axiom,
    ! [Uu: fun_complex_bool,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_cp(Uu),Uua),Uub))
    <=> ( pp(aa_complex_bool(Uu,Uub))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_573
tff(fact_10779_ATP_Olambda__574,axiom,
    ! [Uu: fun_list_nat_bool,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_cl(Uu),Uua),Uub))
    <=> ( pp(aa_list_nat_bool(Uu,Uub))
        & pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_574
tff(fact_10780_ATP_Olambda__575,axiom,
    ! [Uu: fun_set_nat_bool,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_cm(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(Uu,Uub))
        & pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_575
tff(fact_10781_ATP_Olambda__576,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_cn(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_576
tff(fact_10782_ATP_Olambda__577,axiom,
    ! [Uu: fun_int_bool,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_co(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(Uu,Uub))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_577
tff(fact_10783_ATP_Olambda__578,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_aep(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_578
tff(fact_10784_ATP_Olambda__579,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_aes(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_579
tff(fact_10785_ATP_Olambda__580,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_aem(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_580
tff(fact_10786_ATP_Olambda__581,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_aev(Uu),Uua),Uub) = aa_set_int_int(groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_581
tff(fact_10787_ATP_Olambda__582,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_dh(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_582
tff(fact_10788_ATP_Olambda__583,axiom,
    ! [Uu: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_ac(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(Uu,Uub)),Uua) ).

% ATP.lambda_583
tff(fact_10789_ATP_Olambda__584,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_afn(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_584
tff(fact_10790_ATP_Olambda__585,axiom,
    ! [Uu: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_aa(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_option_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_585
tff(fact_10791_ATP_Olambda__586,axiom,
    ! [Uu: fun_bool_assn,Uua: assn,Uub: bool] : aa_bool_assn(aa_ass2376637134808068204l_assn(aTP_Lamm_ab(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(Uu,Uub)),Uua) ).

% ATP.lambda_586
tff(fact_10792_ATP_Olambda__587,axiom,
    ! [Uu: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_ad(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_587
tff(fact_10793_ATP_Olambda__588,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_afo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_588
tff(fact_10794_ATP_Olambda__589,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_afm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_589
tff(fact_10795_ATP_Olambda__590,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_afp(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_590
tff(fact_10796_ATP_Olambda__591,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_nat,Uub: nat] : aa_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_abh(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(Uu,Uub)),Uua) ).

% ATP.lambda_591
tff(fact_10797_ATP_Olambda__592,axiom,
    ! [Uu: fun_real_set_real,Uua: set_real,Uub: real] : aa_real_set_real(aa_set2944350243680128683t_real(aTP_Lamm_yg(Uu),Uua),Uub) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,aa_real_set_real(Uu,Uub)),Uua) ).

% ATP.lambda_592
tff(fact_10798_ATP_Olambda__593,axiom,
    ! [Uu: fun_real_set_bool,Uua: set_bool,Uub: real] : aa_real_set_bool(aa_set5813937368821140009t_bool(aTP_Lamm_yh(Uu),Uua),Uub) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,aa_real_set_bool(Uu,Uub)),Uua) ).

% ATP.lambda_593
tff(fact_10799_ATP_Olambda__594,axiom,
    ! [Uu: fun_real_set_nat,Uua: set_nat,Uub: real] : aa_real_set_nat(aa_set7314861974372065651et_nat(aTP_Lamm_abn(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_real_set_nat(Uu,Uub)),Uua) ).

% ATP.lambda_594
tff(fact_10800_ATP_Olambda__595,axiom,
    ! [Uu: fun_real_set_int,Uua: set_int,Uub: real] : aa_real_set_int(aa_set4051449307867115051et_int(aTP_Lamm_yi(Uu),Uua),Uub) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_real_set_int(Uu,Uub)),Uua) ).

% ATP.lambda_595
tff(fact_10801_ATP_Olambda__596,axiom,
    ! [Uu: fun_bool_set_real,Uua: set_real,Uub: bool] : aa_bool_set_real(aa_set1647705334557766892t_real(aTP_Lamm_yj(Uu),Uua),Uub) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,aa_bool_set_real(Uu,Uub)),Uua) ).

% ATP.lambda_596
tff(fact_10802_ATP_Olambda__597,axiom,
    ! [Uu: fun_bool_set_bool,Uua: set_bool,Uub: bool] : aa_bool_set_bool(aa_set942686091973210910t_bool(aTP_Lamm_yk(Uu),Uua),Uub) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,aa_bool_set_bool(Uu,Uub)),Uua) ).

% ATP.lambda_597
tff(fact_10803_ATP_Olambda__598,axiom,
    ! [Uu: fun_bool_set_nat,Uua: set_nat,Uub: bool] : aa_bool_set_nat(aa_set647555662090937580et_nat(aTP_Lamm_abo(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_bool_set_nat(Uu,Uub)),Uua) ).

% ATP.lambda_598
tff(fact_10804_ATP_Olambda__599,axiom,
    ! [Uu: fun_bool_set_int,Uua: set_int,Uub: bool] : aa_bool_set_int(aa_set1195314601623344620et_int(aTP_Lamm_yl(Uu),Uua),Uub) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_bool_set_int(Uu,Uub)),Uua) ).

% ATP.lambda_599
tff(fact_10805_ATP_Olambda__600,axiom,
    ! [Uu: fun_nat_set_real,Uua: set_real,Uub: nat] : aa_nat_set_real(aa_set5039813165413123151t_real(aTP_Lamm_ym(Uu),Uua),Uub) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,aa_nat_set_real(Uu,Uub)),Uua) ).

% ATP.lambda_600
tff(fact_10806_ATP_Olambda__601,axiom,
    ! [Uu: fun_nat_set_bool,Uua: set_bool,Uub: nat] : aa_nat_set_bool(aa_set6340204957045714437t_bool(aTP_Lamm_yn(Uu),Uua),Uub) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,aa_nat_set_bool(Uu,Uub)),Uua) ).

% ATP.lambda_601
tff(fact_10807_ATP_Olambda__602,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_nat,Uub: nat] : aa_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_abp(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(Uu,Uub)),Uua) ).

% ATP.lambda_602
tff(fact_10808_ATP_Olambda__603,axiom,
    ! [Uu: fun_nat_set_int,Uua: set_int,Uub: nat] : aa_nat_set_int(aa_set2464685511577892303et_int(aTP_Lamm_yo(Uu),Uua),Uub) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_nat_set_int(Uu,Uub)),Uua) ).

% ATP.lambda_603
tff(fact_10809_ATP_Olambda__604,axiom,
    ! [Uu: fun_int_set_real,Uua: set_real,Uub: int] : aa_int_set_real(aa_set1825250779553922219t_real(aTP_Lamm_yp(Uu),Uua),Uub) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,aa_int_set_real(Uu,Uub)),Uua) ).

% ATP.lambda_604
tff(fact_10810_ATP_Olambda__605,axiom,
    ! [Uu: fun_int_set_nat,Uua: set_nat,Uub: int] : aa_int_set_nat(aa_set933959217715569011et_nat(aTP_Lamm_abq(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_int_set_nat(Uu,Uub)),Uua) ).

% ATP.lambda_605
tff(fact_10811_ATP_Olambda__606,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: real,Uub: code_integer] : aa_Code_integer_real(aa_rea8190550742337927152r_real(aTP_Lamm_xu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(Uu,Uub)),Uua) ).

% ATP.lambda_606
tff(fact_10812_ATP_Olambda__607,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: rat,Uub: code_integer] : aa_Code_integer_rat(aa_rat6553461586499541832er_rat(aTP_Lamm_xw(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(Uu,Uub)),Uua) ).

% ATP.lambda_607
tff(fact_10813_ATP_Olambda__608,axiom,
    ! [Uu: fun_complex_real,Uua: real,Uub: complex] : aa_complex_real(aa_rea4562580341872191937x_real(aTP_Lamm_xt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uu,Uub)),Uua) ).

% ATP.lambda_608
tff(fact_10814_ATP_Olambda__609,axiom,
    ! [Uu: fun_complex_rat,Uua: rat,Uub: complex] : aa_complex_rat(aa_rat15117248669498777ex_rat(aTP_Lamm_xv(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_complex_rat(Uu,Uub)),Uua) ).

% ATP.lambda_609
tff(fact_10815_ATP_Olambda__610,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_xx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_610
tff(fact_10816_ATP_Olambda__611,axiom,
    ! [Uu: fun_real_rat,Uua: rat,Uub: real] : aa_real_rat(aa_rat_fun_real_rat(aTP_Lamm_xy(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(Uu,Uub)),Uua) ).

% ATP.lambda_611
tff(fact_10817_ATP_Olambda__612,axiom,
    ! [Uu: fun_bool_real,Uua: real,Uub: bool] : aa_bool_real(aa_rea8194757448443630444l_real(aTP_Lamm_xz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_bool_real(Uu,Uub)),Uua) ).

% ATP.lambda_612
tff(fact_10818_ATP_Olambda__613,axiom,
    ! [Uu: fun_bool_rat,Uua: rat,Uub: bool] : aa_bool_rat(aa_rat_fun_bool_rat(aTP_Lamm_ya(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_bool_rat(Uu,Uub)),Uua) ).

% ATP.lambda_613
tff(fact_10819_ATP_Olambda__614,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_614
tff(fact_10820_ATP_Olambda__615,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_yc(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uub)),Uua) ).

% ATP.lambda_615
tff(fact_10821_ATP_Olambda__616,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aTP_Lamm_jt(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_VEBT_VEBT_bool(Uua,Uub)) ) ) ).

% ATP.lambda_616
tff(fact_10822_ATP_Olambda__617,axiom,
    ! [Uu: complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_jw(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_617
tff(fact_10823_ATP_Olambda__618,axiom,
    ! [Uu: real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_jv(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_618
tff(fact_10824_ATP_Olambda__619,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_jx(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_list_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_619
tff(fact_10825_ATP_Olambda__620,axiom,
    ! [Uu: bool,Uua: fun_bool_bool,Uub: bool] :
      ( pp(aa_bool_bool(aa_fun8098540714523660868l_bool(aTP_Lamm_ju(Uu),Uua),Uub))
    <=> ( ~ ( pp(Uub)
          <=> pp(Uu) )
       => pp(aa_bool_bool(Uua,Uub)) ) ) ).

% ATP.lambda_620
tff(fact_10826_ATP_Olambda__621,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_jy(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_set_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_621
tff(fact_10827_ATP_Olambda__622,axiom,
    ! [Uu: nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_jz(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_622
tff(fact_10828_ATP_Olambda__623,axiom,
    ! [Uu: int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_ka(Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_623
tff(fact_10829_ATP_Olambda__624,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: list_real,Uub: vEBT_VEBT] : aa_VEB6922606862176952244t_real(aa_lis3324575041493669001t_real(aTP_Lamm_aey(Uu),Uua),Uub) = append_real(Uua,cons_real(aa_VEBT_VEBT_real(Uu,Uub),nil_real)) ).

% ATP.lambda_624
tff(fact_10830_ATP_Olambda__625,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: list_nat,Uub: vEBT_VEBT] : aa_VEB4395186035530729816st_nat(aa_lis6183085044976048137st_nat(aTP_Lamm_afb(Uu),Uua),Uub) = append_nat(Uua,aa_list_nat_list_nat(cons_nat(aa_VEBT_VEBT_nat(Uu,Uub)),nil_nat)) ).

% ATP.lambda_625
tff(fact_10831_ATP_Olambda__626,axiom,
    ! [Uu: fun_Pr4298194452408323251l_bool,Uua: list_bool,Uub: produc2804947525362035943l_bool] : aa_Pro3826829133813349938t_bool(aa_lis1035425121888031258t_bool(aTP_Lamm_aez(Uu),Uua),Uub) = append_bool(Uua,cons_bool(aa_Pro9114106320147817132l_bool(Uu,Uub),nil_bool)) ).

% ATP.lambda_626
tff(fact_10832_ATP_Olambda__627,axiom,
    ! [Uu: fun_nat_bool,Uua: list_bool,Uub: nat] : aa_nat_list_bool(aa_lis3589452824828988293t_bool(aTP_Lamm_afa(Uu),Uua),Uub) = append_bool(Uua,cons_bool(aa_nat_bool(Uu,Uub),nil_bool)) ).

% ATP.lambda_627
tff(fact_10833_ATP_Olambda__628,axiom,
    ! [Uu: fun_nat_nat,Uua: list_nat,Uub: nat] : aa_nat_list_nat(aa_lis2277746964813660747st_nat(aTP_Lamm_afc(Uu),Uua),Uub) = append_nat(Uua,aa_list_nat_list_nat(cons_nat(aa_nat_nat(Uu,Uub)),nil_nat)) ).

% ATP.lambda_628
tff(fact_10834_ATP_Olambda__629,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aiu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_629
tff(fact_10835_ATP_Olambda__630,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi,Uub: array_VEBT_VEBTi] : aa_arr8637723718858317150i_assn(aa_VEB7841561841333234298i_assn(aTP_Lamm_px(Uu),Uua),Uub) = snga_assn_VEBT_VEBTi(Uub,replicate_VEBT_VEBTi(Uu,Uua)) ).

% ATP.lambda_630
tff(fact_10836_ATP_Olambda__631,axiom,
    ! [Uu: nat,Uua: bool,Uub: array_bool] : aa_array_bool_assn(aa_boo8717665131543412656l_assn(aTP_Lamm_pw(Uu),Uua),Uub) = snga_assn_bool(Uub,replicate_bool(Uu,Uua)) ).

% ATP.lambda_631
tff(fact_10837_ATP_Olambda__632,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_afr(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_632
tff(fact_10838_ATP_Olambda__633,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_afs(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_633
tff(fact_10839_ATP_Olambda__634,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_afq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_634
tff(fact_10840_ATP_Olambda__635,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aft(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_635
tff(fact_10841_ATP_Olambda__636,axiom,
    ! [Uu: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_ag(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),aa_VEBT_VEBTi_assn(Uu,Uub)) ).

% ATP.lambda_636
tff(fact_10842_ATP_Olambda__637,axiom,
    ! [Uu: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_ae(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),aa_option_nat_assn(Uu,Uub)) ).

% ATP.lambda_637
tff(fact_10843_ATP_Olambda__638,axiom,
    ! [Uu: fun_bool_assn,Uua: assn,Uub: bool] : aa_bool_assn(aa_ass2376637134808068204l_assn(aTP_Lamm_af(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),aa_bool_assn(Uu,Uub)) ).

% ATP.lambda_638
tff(fact_10844_ATP_Olambda__639,axiom,
    ! [Uu: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_ah(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uua),aa_nat_assn(Uu,Uub)) ).

% ATP.lambda_639
tff(fact_10845_ATP_Olambda__640,axiom,
    ! [Uu: set_real,Uua: fun_real_set_real,Uub: real] : aa_real_set_real(aa_fun686715674179075668t_real(aTP_Lamm_yq(Uu),Uua),Uub) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,Uu),aa_real_set_real(Uua,Uub)) ).

% ATP.lambda_640
tff(fact_10846_ATP_Olambda__641,axiom,
    ! [Uu: set_real,Uua: fun_bool_set_real,Uub: bool] : aa_bool_set_real(aa_fun418618139989480252t_real(aTP_Lamm_yt(Uu),Uua),Uub) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,Uu),aa_bool_set_real(Uua,Uub)) ).

% ATP.lambda_641
tff(fact_10847_ATP_Olambda__642,axiom,
    ! [Uu: set_real,Uua: fun_nat_set_real,Uub: nat] : aa_nat_set_real(aa_fun7593858674118267548t_real(aTP_Lamm_yw(Uu),Uua),Uub) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,Uu),aa_nat_set_real(Uua,Uub)) ).

% ATP.lambda_642
tff(fact_10848_ATP_Olambda__643,axiom,
    ! [Uu: set_real,Uua: fun_int_set_real,Uub: int] : aa_int_set_real(aa_fun1830857766034312532t_real(aTP_Lamm_yz(Uu),Uua),Uub) = aa_set_real_set_real(aa_set9029366314755651553t_real(sup_sup_set_real,Uu),aa_int_set_real(Uua,Uub)) ).

% ATP.lambda_643
tff(fact_10849_ATP_Olambda__644,axiom,
    ! [Uu: set_bool,Uua: fun_real_set_bool,Uub: real] : aa_real_set_bool(aa_fun7454438436055979324t_bool(aTP_Lamm_yr(Uu),Uua),Uub) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,Uu),aa_real_set_bool(Uua,Uub)) ).

% ATP.lambda_644
tff(fact_10850_ATP_Olambda__645,axiom,
    ! [Uu: set_bool,Uua: fun_bool_set_bool,Uub: bool] : aa_bool_set_bool(aa_fun2947227189488945156t_bool(aTP_Lamm_yu(Uu),Uua),Uub) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,Uu),aa_bool_set_bool(Uua,Uub)) ).

% ATP.lambda_645
tff(fact_10851_ATP_Olambda__646,axiom,
    ! [Uu: set_bool,Uua: fun_nat_set_bool,Uub: nat] : aa_nat_set_bool(aa_fun4930178584814763708t_bool(aTP_Lamm_yx(Uu),Uua),Uub) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,Uu),aa_nat_set_bool(Uua,Uub)) ).

% ATP.lambda_646
tff(fact_10852_ATP_Olambda__647,axiom,
    ! [Uu: set_nat,Uua: fun_real_set_nat,Uub: real] : aa_real_set_nat(aa_fun8063024115349955228et_nat(aTP_Lamm_abj(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Uu),aa_real_set_nat(Uua,Uub)) ).

% ATP.lambda_647
tff(fact_10853_ATP_Olambda__648,axiom,
    ! [Uu: set_nat,Uua: fun_bool_set_nat,Uub: bool] : aa_bool_set_nat(aa_fun7812420880633473724et_nat(aTP_Lamm_abk(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Uu),aa_bool_set_nat(Uua,Uub)) ).

% ATP.lambda_648
tff(fact_10854_ATP_Olambda__649,axiom,
    ! [Uu: set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_abl(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_649
tff(fact_10855_ATP_Olambda__650,axiom,
    ! [Uu: set_nat,Uua: fun_int_set_nat,Uub: int] : aa_int_set_nat(aa_fun5565777959088800668et_nat(aTP_Lamm_abm(Uu),Uua),Uub) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Uu),aa_int_set_nat(Uua,Uub)) ).

% ATP.lambda_650
tff(fact_10856_ATP_Olambda__651,axiom,
    ! [Uu: set_int,Uua: fun_real_set_int,Uub: real] : aa_real_set_int(aa_fun7998497282572583252et_int(aTP_Lamm_ys(Uu),Uua),Uub) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Uu),aa_real_set_int(Uua,Uub)) ).

% ATP.lambda_651
tff(fact_10857_ATP_Olambda__652,axiom,
    ! [Uu: set_int,Uua: fun_bool_set_int,Uub: bool] : aa_bool_set_int(aa_fun4225254755057748412et_int(aTP_Lamm_yv(Uu),Uua),Uub) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Uu),aa_bool_set_int(Uua,Uub)) ).

% ATP.lambda_652
tff(fact_10858_ATP_Olambda__653,axiom,
    ! [Uu: set_int,Uua: fun_nat_set_int,Uub: nat] : aa_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_yy(Uu),Uua),Uub) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Uu),aa_nat_set_int(Uua,Uub)) ).

% ATP.lambda_653
tff(fact_10859_ATP_Olambda__654,axiom,
    ! [Uu: fun_nat_set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aam(Uu),Uua),Uub))
    <=> pp(aa_set_nat_bool(member_nat(Uub),aa_nat_set_nat(Uu,Uua))) ) ).

% ATP.lambda_654
tff(fact_10860_ATP_Olambda__655,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VE3640887651371881891T_VEBT,Uub: vEBT_VEBT] : aa_VEB8154608084236005916T_VEBT(aa_fun436159583631109252T_VEBT(aTP_Lamm_abd(Uu),Uua),Uub) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Uu),aa_VEB8154608084236005916T_VEBT(Uua,Uub)) ).

% ATP.lambda_655
tff(fact_10861_ATP_Olambda__656,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_re4523157989745374273T_VEBT,Uub: real] : aa_rea5591236203076396202T_VEBT(aa_fun7713332347920806140T_VEBT(aTP_Lamm_za(Uu),Uua),Uub) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Uu),aa_rea5591236203076396202T_VEBT(Uua,Uub)) ).

% ATP.lambda_656
tff(fact_10862_ATP_Olambda__657,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_bo1266566265771860096T_VEBT,Uub: bool] : aa_boo8213446378722761977T_VEBT(aa_fun8832029763029799358T_VEBT(aTP_Lamm_ze(Uu),Uua),Uub) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Uu),aa_boo8213446378722761977T_VEBT(Uua,Uub)) ).

% ATP.lambda_657
tff(fact_10863_ATP_Olambda__658,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_na7453361831206303133T_VEBT,Uub: nat] : aa_nat_set_VEBT_VEBT(aa_fun7681330223414995964T_VEBT(aTP_Lamm_zi(Uu),Uua),Uub) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Uu),aa_nat_set_VEBT_VEBT(Uua,Uub)) ).

% ATP.lambda_658
tff(fact_10864_ATP_Olambda__659,axiom,
    ! [Uu: real,Uua: fun_VE2077312251927228713t_real,Uub: vEBT_VEBT] : aa_VEB5302157898331830618t_real(aa_fun6328685536076230396t_real(aTP_Lamm_abg(Uu),Uua),Uub) = aa_set_real_set_real(insert_real(Uu),aa_VEB5302157898331830618t_real(Uua,Uub)) ).

% ATP.lambda_659
tff(fact_10865_ATP_Olambda__660,axiom,
    ! [Uu: real,Uua: fun_real_set_real,Uub: real] : aa_real_set_real(aa_fun686715674179075668t_real(aTP_Lamm_zb(Uu),Uua),Uub) = aa_set_real_set_real(insert_real(Uu),aa_real_set_real(Uua,Uub)) ).

% ATP.lambda_660
tff(fact_10866_ATP_Olambda__661,axiom,
    ! [Uu: real,Uua: fun_bool_set_real,Uub: bool] : aa_bool_set_real(aa_fun418618139989480252t_real(aTP_Lamm_zf(Uu),Uua),Uub) = aa_set_real_set_real(insert_real(Uu),aa_bool_set_real(Uua,Uub)) ).

% ATP.lambda_661
tff(fact_10867_ATP_Olambda__662,axiom,
    ! [Uu: real,Uua: fun_nat_set_real,Uub: nat] : aa_nat_set_real(aa_fun7593858674118267548t_real(aTP_Lamm_zj(Uu),Uua),Uub) = aa_set_real_set_real(insert_real(Uu),aa_nat_set_real(Uua,Uub)) ).

% ATP.lambda_662
tff(fact_10868_ATP_Olambda__663,axiom,
    ! [Uu: bool,Uua: fun_VE4526288134577429638t_bool,Uub: vEBT_VEBT] : aa_VEB3235182236201257599t_bool(aa_fun2602802399509281482t_bool(aTP_Lamm_abf(Uu),Uua),Uub) = aa_set_bool_set_bool(insert_bool(Uu),aa_VEB3235182236201257599t_bool(Uua,Uub)) ).

% ATP.lambda_663
tff(fact_10869_ATP_Olambda__664,axiom,
    ! [Uu: bool,Uua: fun_real_set_bool,Uub: real] : aa_real_set_bool(aa_fun7454438436055979324t_bool(aTP_Lamm_zc(Uu),Uua),Uub) = aa_set_bool_set_bool(insert_bool(Uu),aa_real_set_bool(Uua,Uub)) ).

% ATP.lambda_664
tff(fact_10870_ATP_Olambda__665,axiom,
    ! [Uu: bool,Uua: fun_bool_set_bool,Uub: bool] : aa_bool_set_bool(aa_fun2947227189488945156t_bool(aTP_Lamm_zg(Uu),Uua),Uub) = aa_set_bool_set_bool(insert_bool(Uu),aa_bool_set_bool(Uua,Uub)) ).

% ATP.lambda_665
tff(fact_10871_ATP_Olambda__666,axiom,
    ! [Uu: bool,Uua: fun_nat_set_bool,Uub: nat] : aa_nat_set_bool(aa_fun4930178584814763708t_bool(aTP_Lamm_abc(Uu),Uua),Uub) = aa_set_bool_set_bool(insert_bool(Uu),aa_nat_set_bool(Uua,Uub)) ).

% ATP.lambda_666
tff(fact_10872_ATP_Olambda__667,axiom,
    ! [Uu: int,Uua: fun_VE6538172513902245929et_int,Uub: vEBT_VEBT] : aa_VEBT_VEBT_set_int(aa_fun7169446219367245564et_int(aTP_Lamm_abe(Uu),Uua),Uub) = aa_set_int_set_int(insert_int(Uu),aa_VEBT_VEBT_set_int(Uua,Uub)) ).

% ATP.lambda_667
tff(fact_10873_ATP_Olambda__668,axiom,
    ! [Uu: int,Uua: fun_real_set_int,Uub: real] : aa_real_set_int(aa_fun7998497282572583252et_int(aTP_Lamm_zd(Uu),Uua),Uub) = aa_set_int_set_int(insert_int(Uu),aa_real_set_int(Uua,Uub)) ).

% ATP.lambda_668
tff(fact_10874_ATP_Olambda__669,axiom,
    ! [Uu: int,Uua: fun_bool_set_int,Uub: bool] : aa_bool_set_int(aa_fun4225254755057748412et_int(aTP_Lamm_zh(Uu),Uua),Uub) = aa_set_int_set_int(insert_int(Uu),aa_bool_set_int(Uua,Uub)) ).

% ATP.lambda_669
tff(fact_10875_ATP_Olambda__670,axiom,
    ! [Uu: int,Uua: fun_nat_set_int,Uub: nat] : aa_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_abb(Uu),Uua),Uub) = aa_set_int_set_int(insert_int(Uu),aa_nat_set_int(Uua,Uub)) ).

% ATP.lambda_670
tff(fact_10876_ATP_Olambda__671,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_real(aa_fun6014602394608682688t_real(aTP_Lamm_aax(Uu),Uua),Uub) = aa_set_nat_set_real(image_nat_real(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_671
tff(fact_10877_ATP_Olambda__672,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_set_nat(aa_fun3336736533861547162et_nat(aTP_Lamm_aay(Uu),Uua),Uub) = aa_set7246812951530217472et_nat(image_nat_set_nat(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_672
tff(fact_10878_ATP_Olambda__673,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aba(Uu),Uua),Uub) = aa_set_nat_set_nat(image_nat_nat(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_673
tff(fact_10879_ATP_Olambda__674,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_int(aa_fun7583645416216832320et_int(aTP_Lamm_aaz(Uu),Uua),Uub) = aa_set_nat_set_int(image_nat_int(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_674
tff(fact_10880_ATP_Olambda__675,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(aTP_Lamm_ce(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),Uua)),aa_nat_nat(suc,aa_Code_integer_nat(Uu,Uub))),aa_Code_integer_nat(Uu,Uub)))) ) ).

% ATP.lambda_675
tff(fact_10881_ATP_Olambda__676,axiom,
    ! [Uu: fun_complex_nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_cd(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uua)),aa_nat_nat(suc,aa_complex_nat(Uu,Uub))),aa_complex_nat(Uu,Uub)))) ) ).

% ATP.lambda_676
tff(fact_10882_ATP_Olambda__677,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(aTP_Lamm_bz(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,Uub),Uua)),aa_nat_nat(suc,aa_list_nat_nat(Uu,Uub))),aa_list_nat_nat(Uu,Uub)))) ) ).

% ATP.lambda_677
tff(fact_10883_ATP_Olambda__678,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(aTP_Lamm_ca(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,Uub),Uua)),aa_nat_nat(suc,aa_set_nat_nat(Uu,Uub))),aa_set_nat_nat(Uu,Uub)))) ) ).

% ATP.lambda_678
tff(fact_10884_ATP_Olambda__679,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cb(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uua)),aa_nat_nat(suc,aa_nat_nat(Uu,Uub))),aa_nat_nat(Uu,Uub)))) ) ).

% ATP.lambda_679
tff(fact_10885_ATP_Olambda__680,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_cc(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uua)),aa_nat_nat(suc,aa_int_nat(Uu,Uub))),aa_int_nat(Uu,Uub)))) ) ).

% ATP.lambda_680
tff(fact_10886_ATP_Olambda__681,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Co8822979462182099027r_bool,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun8875953811088571772r_bool(aTP_Lamm_by(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(Uua,Uub)),aa_Code_integer_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_681
tff(fact_10887_ATP_Olambda__682,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_bw(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(Uua,Uub)),aa_complex_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_682
tff(fact_10888_ATP_Olambda__683,axiom,
    ! [Uu: fun_list_nat_nat,Uua: fun_list_nat_bool,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun6009467934925565500t_bool(aTP_Lamm_bo(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_list_nat_bool(Uua,Uub)),aa_list_nat_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_683
tff(fact_10889_ATP_Olambda__684,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_bool,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun3565389549526402364t_bool(aTP_Lamm_bq(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_set_nat_bool(Uua,Uub)),aa_set_nat_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_684
tff(fact_10890_ATP_Olambda__685,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_bs(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(Uua,Uub)),aa_nat_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_685
tff(fact_10891_ATP_Olambda__686,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_bu(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(Uua,Uub)),aa_int_nat(Uu,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_686
tff(fact_10892_ATP_Olambda__687,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun3540204951715082475r_bool(aTP_Lamm_hp(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uua,Uub)))) ) ).

% ATP.lambda_687
tff(fact_10893_ATP_Olambda__688,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun4400319914713427657x_bool(aTP_Lamm_ho(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uua,Uub)))) ) ).

% ATP.lambda_688
tff(fact_10894_ATP_Olambda__689,axiom,
    ! [Uu: fun_list_nat_nat,Uua: fun_list_nat_nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_fun2581403542315806821t_bool(aTP_Lamm_hk(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_nat_nat(Uu,Uub)),aa_list_nat_nat(Uua,Uub)))) ) ).

% ATP.lambda_689
tff(fact_10895_ATP_Olambda__690,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_fun806690937814068505t_bool(aTP_Lamm_hl(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uu,Uub)),aa_set_nat_nat(Uua,Uub)))) ) ).

% ATP.lambda_690
tff(fact_10896_ATP_Olambda__691,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1783405840085286277t_bool(aTP_Lamm_hm(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)))) ) ).

% ATP.lambda_691
tff(fact_10897_ATP_Olambda__692,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( pp(aa_int_bool(aa_fun5627573752281863885t_bool(aTP_Lamm_hn(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub)))) ) ).

% ATP.lambda_692
tff(fact_10898_ATP_Olambda__693,axiom,
    ! [Uu: fun_real_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_th(Uu),Uua),Uub) = aa_real_real(Uu,aa_nat_real(Uua,Uub)) ).

% ATP.lambda_693
tff(fact_10899_ATP_Olambda__694,axiom,
    ! [Uu: fun_real_set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_set_nat(aa_fun7471991306178591626et_nat(aTP_Lamm_aat(Uu),Uua),Uub) = aa_real_set_nat(Uu,aa_nat_real(Uua,Uub)) ).

% ATP.lambda_694
tff(fact_10900_ATP_Olambda__695,axiom,
    ! [Uu: fun_real_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_nat(aa_fun8286852990929645268at_nat(aTP_Lamm_ti(Uu),Uua),Uub) = aa_real_nat(Uu,aa_nat_real(Uua,Uub)) ).

% ATP.lambda_695
tff(fact_10901_ATP_Olambda__696,axiom,
    ! [Uu: fun_real_int,Uua: fun_nat_real,Uub: nat] : aa_nat_int(aa_fun4109001971420448560at_int(aTP_Lamm_tj(Uu),Uua),Uub) = aa_real_int(Uu,aa_nat_real(Uua,Uub)) ).

% ATP.lambda_696
tff(fact_10902_ATP_Olambda__697,axiom,
    ! [Uu: fun_set_nat_set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aau(Uu),Uua),Uub) = aa_set_nat_set_nat(Uu,aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_697
tff(fact_10903_ATP_Olambda__698,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_tm(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_698
tff(fact_10904_ATP_Olambda__699,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_set_nat(aa_fun7909995953855020590et_nat(aTP_Lamm_aaw(Uu),Uua),Uub) = aa_nat_set_nat(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_699
tff(fact_10905_ATP_Olambda__700,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tn(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_700
tff(fact_10906_ATP_Olambda__701,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_tp(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_701
tff(fact_10907_ATP_Olambda__702,axiom,
    ! [Uu: fun_nat_int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_to(Uu),Uua),Uub) = aa_nat_int(Uu,aa_int_nat(Uua,Uub)) ).

% ATP.lambda_702
tff(fact_10908_ATP_Olambda__703,axiom,
    ! [Uu: fun_int_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_tk(Uu),Uua),Uub) = aa_int_real(Uu,aa_nat_int(Uua,Uub)) ).

% ATP.lambda_703
tff(fact_10909_ATP_Olambda__704,axiom,
    ! [Uu: fun_int_set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_set_nat(aa_fun3079086916031139466et_nat(aTP_Lamm_aav(Uu),Uua),Uub) = aa_int_set_nat(Uu,aa_nat_int(Uua,Uub)) ).

% ATP.lambda_704
tff(fact_10910_ATP_Olambda__705,axiom,
    ! [Uu: fun_int_set_nat,Uua: fun_int_int,Uub: int] : aa_int_set_nat(aa_fun2608467900932457282et_nat(aTP_Lamm_aas(Uu),Uua),Uub) = aa_int_set_nat(Uu,aa_int_int(Uua,Uub)) ).

% ATP.lambda_705
tff(fact_10911_ATP_Olambda__706,axiom,
    ! [Uu: fun_int_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_nat(aa_fun519522104774525396at_nat(aTP_Lamm_tl(Uu),Uua),Uub) = aa_int_nat(Uu,aa_nat_int(Uua,Uub)) ).

% ATP.lambda_706
tff(fact_10912_ATP_Olambda__707,axiom,
    ! [Uu: fun_int_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_tq(Uu),Uua),Uub) = aa_int_int(Uu,aa_nat_int(Uua,Uub)) ).

% ATP.lambda_707
tff(fact_10913_ATP_Olambda__708,axiom,
    ! [Uu: fun_na6128946301132549280et_nat,Uua: set_nat,Uub: nat] : aa_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_aap(Uu),Uua),Uub) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_aao(Uu),Uub)),Uua)) ).

% ATP.lambda_708
tff(fact_10914_ATP_Olambda__709,axiom,
    ! [Uu: fun_na6128946301132549280et_nat,Uua: set_nat,Uub: nat] : aa_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_aan(Uu),Uua),Uub) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aa_nat1818925681583257441et_nat(Uu,Uub)),Uua)) ).

% ATP.lambda_709
tff(fact_10915_ATP_Olambda__710,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_aix(Uu),Uua),Uub) = aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),Uua))) ).

% ATP.lambda_710
tff(fact_10916_ATP_Olambda__711,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_aiw(Uu),Uua),Uub) = aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),Uua))) ).

% ATP.lambda_711
tff(fact_10917_ATP_Olambda__712,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_aaq(Uu),Uua),Uub) = comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(Uu),aa_nat_set_nat(Uua,Uub))) ).

% ATP.lambda_712
tff(fact_10918_ATP_Olambda__713,axiom,
    ! [Uu: fun_re1255401033344146673l_bool,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aa_fun2912881648665429622T_bool(aTP_Lamm_hc(Uu),Uua),Uub),Uuc))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(Uu,aa_VEBT_VEBT_real(Uua,Uub)),aa_VEBT_VEBT_real(Uua,Uuc))) ) ).

% ATP.lambda_713
tff(fact_10919_ATP_Olambda__714,axiom,
    ! [Uu: fun_bo1841576699660343469l_bool,Uua: fun_Pr4298194452408323251l_bool,Uub: produc2804947525362035943l_bool,Uuc: produc2804947525362035943l_bool] :
      ( pp(aa_Pro9114106320147817132l_bool(aa_Pro6708868278645019742l_bool(aa_fun4430674632827408636l_bool(aTP_Lamm_hd(Uu),Uua),Uub),Uuc))
    <=> pp(aa_bool_bool(aa_boo8970175072276269406l_bool(Uu,aa_Pro9114106320147817132l_bool(Uua,Uub)),aa_Pro9114106320147817132l_bool(Uua,Uuc))) ) ).

% ATP.lambda_714
tff(fact_10920_ATP_Olambda__715,axiom,
    ! [Uu: fun_bo1841576699660343469l_bool,Uua: fun_nat_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun8271680885375493061t_bool(aTP_Lamm_he(Uu),Uua),Uub),Uuc))
    <=> pp(aa_bool_bool(aa_boo8970175072276269406l_bool(Uu,aa_nat_bool(Uua,Uub)),aa_nat_bool(Uua,Uuc))) ) ).

% ATP.lambda_715
tff(fact_10921_ATP_Olambda__716,axiom,
    ! [Uu: fun_nat_fun_nat_bool,Uua: fun_VEBT_VEBT_nat,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aa_fun7053968375109723858T_bool(aTP_Lamm_hf(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(Uu,aa_VEBT_VEBT_nat(Uua,Uub)),aa_VEBT_VEBT_nat(Uua,Uuc))) ) ).

% ATP.lambda_716
tff(fact_10922_ATP_Olambda__717,axiom,
    ! [Uu: fun_nat_fun_nat_bool,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3777555276967886990t_bool(aTP_Lamm_hg(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(Uu,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uua,Uuc))) ) ).

% ATP.lambda_717
tff(fact_10923_ATP_Olambda__718,axiom,
    ! [Uu: fun_nat_fun_nat_bool,Uua: fun_int_nat,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_fun2912556895731619962t_bool(aTP_Lamm_hh(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(Uu,aa_int_nat(Uua,Uub)),aa_int_nat(Uua,Uuc))) ) ).

% ATP.lambda_718
tff(fact_10924_ATP_Olambda__719,axiom,
    ! [Uu: fun_int_fun_int_bool,Uua: fun_nat_int,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun2605284095902351154t_bool(aTP_Lamm_hi(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(Uu,aa_nat_int(Uua,Uub)),aa_nat_int(Uua,Uuc))) ) ).

% ATP.lambda_719
tff(fact_10925_ATP_Olambda__720,axiom,
    ! [Uu: fun_int_fun_int_bool,Uua: fun_int_int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_fun1740285714666084126t_bool(aTP_Lamm_hj(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(Uu,aa_int_int(Uua,Uub)),aa_int_int(Uua,Uuc))) ) ).

% ATP.lambda_720
tff(fact_10926_ATP_Olambda__721,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_afl(Uu),Uua),Uub),Uuc) = aa_set_int_int(groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_afk(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_721
tff(fact_10927_ATP_Olambda__722,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_afj(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_afi(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_722
tff(fact_10928_ATP_Olambda__723,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_afh(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_afg(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_723
tff(fact_10929_ATP_Olambda__724,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_aff(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_afe(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_724
tff(fact_10930_ATP_Olambda__725,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co7124838579184476414x_bool,Uuc: complex] : aa_complex_complex(aa_fun8169706760919072411omplex(aa_fun2334956106851614393omplex(aTP_Lamm_aht(Uu),Uua),Uub),Uuc) = groups8024822376189712711omplex(aa_com7552888802784664052omplex(aTP_Lamm_ahr(Uua),Uuc),collect_Code_integer(aa_com339898768654300263r_bool(aa_fun8191597921216526780r_bool(aTP_Lamm_ahs(Uu),Uub),Uuc))) ).

% ATP.lambda_725
tff(fact_10931_ATP_Olambda__726,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: complex] : aa_complex_complex(aa_fun2686559102040213132omplex(aa_fun5422379709423809691omplex(aTP_Lamm_aho(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_ahm(Uua),Uuc),collect_int(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_ahn(Uu),Uub),Uuc))) ).

% ATP.lambda_726
tff(fact_10932_ATP_Olambda__727,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: complex] : aa_complex_complex(aa_fun1613615970879741416omplex(aa_fun7586814990830484307omplex(aTP_Lamm_ahj(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ahh(Uua),Uuc),collect_nat(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_ahi(Uu),Uub),Uuc))) ).

% ATP.lambda_727
tff(fact_10933_ATP_Olambda__728,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: complex] : aa_complex_complex(aa_fun8128191506076533516omplex(aa_fun840110154952133019omplex(aTP_Lamm_ahe(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_ahd(Uua),Uuc),collect_real(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_dc(Uu),Uub),Uuc))) ).

% ATP.lambda_728
tff(fact_10934_ATP_Olambda__729,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE2128610149277289041x_bool,Uuc: complex] : aa_complex_complex(aa_fun1095524714264483688omplex(aa_fun8876071380179997707omplex(aTP_Lamm_aha(Uu),Uua),Uub),Uuc) = groups1794756597179926696omplex(aa_com2862003112617944841omplex(aTP_Lamm_agz(Uua),Uuc),collect_VEBT_VEBT(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_db(Uu),Uub),Uuc))) ).

% ATP.lambda_729
tff(fact_10935_ATP_Olambda__730,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co8657364630979331932t_bool,Uuc: nat] : aa_nat_nat(aa_fun7549014261834767933at_nat(aa_fun5721305349739569687at_nat(aTP_Lamm_agw(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_agu(Uua),Uuc),collect_Code_integer(aa_nat6120813132205607877r_bool(aa_fun8839232882156773436r_bool(aTP_Lamm_agv(Uu),Uub),Uuc))) ).

% ATP.lambda_730
tff(fact_10936_ATP_Olambda__731,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: nat] : aa_nat_nat(aa_fun996456248398975532at_nat(aa_fun8754082201773474805at_nat(aTP_Lamm_agr(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_agp(Uua),Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_agq(Uu),Uub),Uuc))) ).

% ATP.lambda_731
tff(fact_10937_ATP_Olambda__732,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: nat] : aa_nat_nat(aa_fun4133023758092209198at_nat(aa_fun7384697781603784697at_nat(aTP_Lamm_agm(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_agk(Uua),Uuc),collect_int(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_agl(Uu),Uub),Uuc))) ).

% ATP.lambda_732
tff(fact_10938_ATP_Olambda__733,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: nat] : aa_nat_nat(aa_fun5672750458368166062at_nat(aa_fun6824837968375615225at_nat(aTP_Lamm_agh(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_agg(Uua),Uuc),collect_real(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_cy(Uu),Uub),Uuc))) ).

% ATP.lambda_733
tff(fact_10939_ATP_Olambda__734,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: nat] : aa_nat_nat(aa_fun5562694175296226570at_nat(aa_fun8318684123515291377at_nat(aTP_Lamm_agd(Uu),Uua),Uub),Uuc) = groups771621172384141258BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_agc(Uua),Uuc),collect_VEBT_VEBT(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_cx(Uu),Uub),Uuc))) ).

% ATP.lambda_734
tff(fact_10940_ATP_Olambda__735,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_aih(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_aig(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_735
tff(fact_10941_ATP_Olambda__736,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ais(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_736
tff(fact_10942_ATP_Olambda__737,axiom,
    ! [Uu: fun_VE3165209603783583043T_VEBT,Uua: set_VEBT_VEBT,Uub: fun_VEBT_VEBT_bool,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aa_set5278205761110950305T_bool(aTP_Lamm_tr(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),image_3375948659692109573T_VEBT(Uu,Uua)))
        & pp(aa_VEBT_VEBT_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_737
tff(fact_10943_ATP_Olambda__738,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: set_VEBT_VEBT,Uub: fun_real_bool,Uuc: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aa_set4098556036960817563l_bool(aTP_Lamm_tv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),image_VEBT_VEBT_real(Uu,Uua)))
        & pp(aa_real_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_738
tff(fact_10944_ATP_Olambda__739,axiom,
    ! [Uu: fun_co4340422964966740447T_VEBT,Uua: set_complex,Uub: fun_VEBT_VEBT_bool,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aa_set1540900332140587627T_bool(aTP_Lamm_tz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),image_932796090930683071T_VEBT(Uu,Uua)))
        & pp(aa_VEBT_VEBT_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_739
tff(fact_10945_ATP_Olambda__740,axiom,
    ! [Uu: fun_complex_real,Uua: set_complex,Uub: fun_real_bool,Uuc: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aa_set6829441243277608293l_bool(aTP_Lamm_ub(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),image_complex_real(Uu,Uua)))
        & pp(aa_real_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_740
tff(fact_10946_ATP_Olambda__741,axiom,
    ! [Uu: fun_real_VEBT_VEBT,Uua: set_real,Uub: fun_VEBT_VEBT_bool,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aa_set6789171798632725353T_bool(aTP_Lamm_tt(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),image_real_VEBT_VEBT(Uu,Uua)))
        & pp(aa_VEBT_VEBT_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_741
tff(fact_10947_ATP_Olambda__742,axiom,
    ! [Uu: fun_real_real,Uua: set_real,Uub: fun_real_bool,Uuc: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aa_set1278274404314589027l_bool(aTP_Lamm_tx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),image_real_real(Uu,Uua)))
        & pp(aa_real_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_742
tff(fact_10948_ATP_Olambda__743,axiom,
    ! [Uu: fun_nat_VEBT_VEBT,Uua: set_nat,Uub: fun_VEBT_VEBT_bool,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aa_set5197589798974504589T_bool(aTP_Lamm_ud(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),image_nat_VEBT_VEBT(Uu,Uua)))
        & pp(aa_VEBT_VEBT_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_743
tff(fact_10949_ATP_Olambda__744,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_real_bool,Uuc: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aa_set512738241545386119l_bool(aTP_Lamm_uf(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),aa_set_nat_set_real(image_nat_real(Uu),Uua)))
        & pp(aa_real_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_744
tff(fact_10950_ATP_Olambda__745,axiom,
    ! [Uu: fun_int_VEBT_VEBT,Uua: set_int,Uub: fun_VEBT_VEBT_bool,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aa_set6809977906677068521T_bool(aTP_Lamm_uh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),image_int_VEBT_VEBT(Uu,Uua)))
        & pp(aa_VEBT_VEBT_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_745
tff(fact_10951_ATP_Olambda__746,axiom,
    ! [Uu: fun_int_real,Uua: set_int,Uub: fun_real_bool,Uuc: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aa_set8393313469388136163l_bool(aTP_Lamm_uj(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),image_int_real(Uu,Uua)))
        & pp(aa_real_bool(Uub,Uuc)) ) ) ).

% ATP.lambda_746
tff(fact_10952_ATP_Olambda__747,axiom,
    ! [Uu: set_complex,Uua: fun_VE2128610149277289041x_bool,Uub: vEBT_VEBT,Uuc: complex] :
      ( pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(aa_fun5298830385910147808x_bool(aTP_Lamm_agx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_747
tff(fact_10953_ATP_Olambda__748,axiom,
    ! [Uu: set_complex,Uua: fun_Co7124838579184476414x_bool,Uub: code_integer,Uuc: complex] :
      ( pp(aa_complex_bool(aa_Cod6819470024986797159x_bool(aa_fun3619973020723243580x_bool(aTP_Lamm_ahp(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_Cod6819470024986797159x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_748
tff(fact_10954_ATP_Olambda__749,axiom,
    ! [Uu: set_complex,Uua: fun_re3010880487650781423x_bool,Uub: real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_ahb(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_749
tff(fact_10955_ATP_Olambda__750,axiom,
    ! [Uu: set_complex,Uua: fun_na5941084329111710283x_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_ahf(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_750
tff(fact_10956_ATP_Olambda__751,axiom,
    ! [Uu: set_complex,Uua: fun_in5750378413879446255x_bool,Uub: int,Uuc: complex] :
      ( pp(aa_complex_bool(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_ahk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_751
tff(fact_10957_ATP_Olambda__752,axiom,
    ! [Uu: set_nat,Uua: fun_VE3763956218935105199t_bool,Uub: vEBT_VEBT,Uuc: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_aga(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_752
tff(fact_10958_ATP_Olambda__753,axiom,
    ! [Uu: set_nat,Uua: fun_Co8657364630979331932t_bool,Uub: code_integer,Uuc: nat] :
      ( pp(aa_nat_bool(aa_Cod2498268374540730821t_bool(aa_fun498830667440607164t_bool(aTP_Lamm_ags(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_Cod2498268374540730821t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_753
tff(fact_10959_ATP_Olambda__754,axiom,
    ! [Uu: set_nat,Uua: fun_co2841405903858895691t_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_agn(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_754
tff(fact_10960_ATP_Olambda__755,axiom,
    ! [Uu: set_nat,Uua: fun_re9121486324101951821t_bool,Uub: real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_age(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_755
tff(fact_10961_ATP_Olambda__756,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_agi(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_756
tff(fact_10962_ATP_Olambda__757,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE1123255499321568226r_bool,Uub: code_integer,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_Cod3526525079615536833T_bool(aa_fun7592865282689444384T_bool(aTP_Lamm_dd(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & pp(aa_Code_integer_bool(aa_VEB6607917098650726363r_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_757
tff(fact_10963_ATP_Olambda__758,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2128610149277289041x_bool,Uub: complex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_com4769948546659418770T_bool(aa_fun3341029659329701088T_bool(aTP_Lamm_db(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & pp(aa_complex_bool(aa_VEB8450443329232412106x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_758
tff(fact_10964_ATP_Olambda__759,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE3763956218935105199t_bool,Uub: nat,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(aa_fun4018725027000996448T_bool(aTP_Lamm_cx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_759
tff(fact_10965_ATP_Olambda__760,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE3303673267288458579t_bool,Uub: int,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_int3199078826956362512T_bool(aa_fun4417922564241321824T_bool(aTP_Lamm_cz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & pp(aa_int_bool(aa_VEB1707969225601237452t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_760
tff(fact_10966_ATP_Olambda__761,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co7124838579184476414x_bool,Uub: complex,Uuc: code_integer] :
      ( pp(aa_Code_integer_bool(aa_com339898768654300263r_bool(aa_fun8191597921216526780r_bool(aTP_Lamm_ahs(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uuc),Uu))
        & pp(aa_complex_bool(aa_Cod6819470024986797159x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_761
tff(fact_10967_ATP_Olambda__762,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8657364630979331932t_bool,Uub: nat,Uuc: code_integer] :
      ( pp(aa_Code_integer_bool(aa_nat6120813132205607877r_bool(aa_fun8839232882156773436r_bool(aTP_Lamm_agv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uuc),Uu))
        & pp(aa_nat_bool(aa_Cod2498268374540730821t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_762
tff(fact_10968_ATP_Olambda__763,axiom,
    ! [Uu: set_complex,Uua: fun_co2841405903858895691t_bool,Uub: nat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aa_fun5091585579174883964x_bool(aTP_Lamm_agq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & pp(aa_nat_bool(aa_com5335841084339320244t_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_763
tff(fact_10969_ATP_Olambda__764,axiom,
    ! [Uu: set_real,Uua: fun_re1120004649140997760r_bool,Uub: code_integer,Uuc: real] :
      ( pp(aa_real_bool(aa_Cod181991726960451049l_bool(aa_fun8452463775060891964l_bool(aTP_Lamm_de(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_Code_integer_bool(aa_rea7928672538722535017r_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_764
tff(fact_10970_ATP_Olambda__765,axiom,
    ! [Uu: set_real,Uua: fun_re3010880487650781423x_bool,Uub: complex,Uuc: real] :
      ( pp(aa_real_bool(aa_com8943241093633526360l_bool(aa_fun8455611678790336124l_bool(aTP_Lamm_dc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_complex_bool(aa_rea5887071448072802392x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_765
tff(fact_10971_ATP_Olambda__766,axiom,
    ! [Uu: set_real,Uua: fun_re9121486324101951821t_bool,Uub: nat,Uuc: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aa_fun3382945097342017788l_bool(aTP_Lamm_cy(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_nat_bool(aa_real_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_766
tff(fact_10972_ATP_Olambda__767,axiom,
    ! [Uu: set_real,Uua: fun_re8661203372455305201t_bool,Uub: int,Uuc: real] :
      ( pp(aa_real_bool(aa_int_fun_real_bool(aa_fun1536421850685375484l_bool(aTP_Lamm_da(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & pp(aa_int_bool(aa_real_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_767
tff(fact_10973_ATP_Olambda__768,axiom,
    ! [Uu: set_nat,Uua: fun_na5941084329111710283x_bool,Uub: complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_com5335841084339320244t_bool(aa_fun8674784841953735804t_bool(aTP_Lamm_ahi(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_complex_bool(aa_nat5545486172511849908x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_768
tff(fact_10974_ATP_Olambda__769,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_bool,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3051233943552222972t_bool(aTP_Lamm_df(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_769
tff(fact_10975_ATP_Olambda__770,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_bool,Uub: int,Uuc: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(aa_fun8184653438340336124t_bool(aTP_Lamm_dg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & pp(aa_int_bool(aa_nat_fun_int_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_770
tff(fact_10976_ATP_Olambda__771,axiom,
    ! [Uu: set_int,Uua: fun_in5750378413879446255x_bool,Uub: complex,Uuc: int] :
      ( pp(aa_int_bool(aa_com2211550117870696024t_bool(aa_fun3087709329828230012t_bool(aTP_Lamm_ahn(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_complex_bool(aa_int5354780257279585880x_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_771
tff(fact_10977_ATP_Olambda__772,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_bool,Uub: nat,Uuc: int] :
      ( pp(aa_int_bool(aa_nat_fun_int_bool(aa_fun3035398748895483388t_bool(aTP_Lamm_agl(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_772
tff(fact_10978_ATP_Olambda__773,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE9219535225822561060nteger,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aa_fun1850620383256809093T_bool(aTP_Lamm_ur(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_VEB5935388676523804373nteger(Uua,Uuc) = aa_VEB5935388676523804373nteger(Uua,Uub) ) ) ) ).

% ATP.lambda_773
tff(fact_10979_ATP_Olambda__774,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aa_fun5117077068891053428T_bool(aTP_Lamm_up(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_VEBT_VEBT_complex(Uua,Uuc) = aa_VEBT_VEBT_complex(Uua,Uub) ) ) ) ).

% ATP.lambda_774
tff(fact_10980_ATP_Olambda__775,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aa_fun7053968375109723858T_bool(aTP_Lamm_ul(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_VEBT_VEBT_nat(Uua,Uuc) = aa_VEBT_VEBT_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_775
tff(fact_10981_ATP_Olambda__776,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_int,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(aa_fun3060702913319154550T_bool(aTP_Lamm_un(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_VEBT_VEBT_int(Uua,Uuc) = aa_VEBT_VEBT_int(Uua,Uub) ) ) ) ).

% ATP.lambda_776
tff(fact_10982_ATP_Olambda__777,axiom,
    ! [Uu: set_real,Uua: fun_re6978948838062957830nteger,Uub: real,Uuc: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aa_fun5856998839660973357l_bool(aTP_Lamm_us(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_Code_integer(Uua,Uuc) = aa_real_Code_integer(Uua,Uub) ) ) ) ).

% ATP.lambda_777
tff(fact_10983_ATP_Olambda__778,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real,Uuc: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aa_fun3442657443821130908l_bool(aTP_Lamm_uq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_complex(Uua,Uuc) = aa_real_complex(Uua,Uub) ) ) ) ).

% ATP.lambda_778
tff(fact_10984_ATP_Olambda__779,axiom,
    ! [Uu: set_real,Uua: fun_real_nat,Uub: real,Uuc: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aa_fun3203339554537555706l_bool(aTP_Lamm_um(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_nat(Uua,Uuc) = aa_real_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_779
tff(fact_10985_ATP_Olambda__780,axiom,
    ! [Uu: set_real,Uua: fun_real_int,Uub: real,Uuc: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aa_fun7220475794303443358l_bool(aTP_Lamm_uo(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_int(Uua,Uuc) = aa_real_int(Uua,Uub) ) ) ) ).

% ATP.lambda_780
tff(fact_10986_ATP_Olambda__781,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun2005584043565060658t_bool(aTP_Lamm_ut(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_nat_real(Uua,Uuc) = aa_nat_real(Uua,Uub) ) ) ) ).

% ATP.lambda_781
tff(fact_10987_ATP_Olambda__782,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_fun3777555276967886990t_bool(aTP_Lamm_uu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_nat_nat(Uua,Uuc) = aa_nat_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_782
tff(fact_10988_ATP_Olambda__783,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_air(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_783
tff(fact_10989_ATP_Olambda__784,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_akx(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_784
tff(fact_10990_ATP_Olambda__785,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_aku(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_785
tff(fact_10991_ATP_Olambda__786,axiom,
    ! [Uu: fun_VE3165209603783583043T_VEBT,Uua: set_VEBT_VEBT,Uub: fun_VEBT_VEBT_bool,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aa_set5278205761110950305T_bool(aTP_Lamm_ts(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uua))
        & pp(aa_VEBT_VEBT_bool(Uub,aa_VEB5832953974749077948T_VEBT(Uu,Uuc))) ) ) ).

% ATP.lambda_786
tff(fact_10992_ATP_Olambda__787,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: set_VEBT_VEBT,Uub: fun_real_bool,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun2927746532475200174T_bool(aa_set655613085781954469T_bool(aTP_Lamm_tw(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uua))
        & pp(aa_real_bool(Uub,aa_VEBT_VEBT_real(Uu,Uuc))) ) ) ).

% ATP.lambda_787
tff(fact_10993_ATP_Olambda__788,axiom,
    ! [Uu: fun_co4340422964966740447T_VEBT,Uua: set_complex,Uub: fun_VEBT_VEBT_bool,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun3721220998789367262x_bool(aa_set4710227632152643155x_bool(aTP_Lamm_ua(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uua))
        & pp(aa_VEBT_VEBT_bool(Uub,aa_complex_VEBT_VEBT(Uu,Uuc))) ) ) ).

% ATP.lambda_788
tff(fact_10994_ATP_Olambda__789,axiom,
    ! [Uu: fun_complex_real,Uua: set_complex,Uub: fun_real_bool,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun3481569711432096570x_bool(aa_set768065858760106851x_bool(aTP_Lamm_uc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uua))
        & pp(aa_real_bool(Uub,aa_complex_real(Uu,Uuc))) ) ) ).

% ATP.lambda_789
tff(fact_10995_ATP_Olambda__790,axiom,
    ! [Uu: fun_real_VEBT_VEBT,Uua: set_real,Uub: fun_VEBT_VEBT_bool,Uuc: real] :
      ( pp(aa_real_bool(aa_fun2240483520789045728l_bool(aa_set4656197680294135511l_bool(aTP_Lamm_tu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uua))
        & pp(aa_VEBT_VEBT_bool(Uub,aa_real_VEBT_VEBT(Uu,Uuc))) ) ) ).

% ATP.lambda_790
tff(fact_10996_ATP_Olambda__791,axiom,
    ! [Uu: fun_real_real,Uua: set_real,Uub: fun_real_bool,Uuc: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aa_set1278274404314589027l_bool(aTP_Lamm_ty(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uua))
        & pp(aa_real_bool(Uub,aa_real_real(Uu,Uuc))) ) ) ).

% ATP.lambda_791
tff(fact_10997_ATP_Olambda__792,axiom,
    ! [Uu: fun_nat_VEBT_VEBT,Uua: set_nat,Uub: fun_VEBT_VEBT_bool,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun8987042180644983100t_bool(aa_set2270706540308894223t_bool(aTP_Lamm_ue(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uua))
        & pp(aa_VEBT_VEBT_bool(Uub,aa_nat_VEBT_VEBT(Uu,Uuc))) ) ) ).

% ATP.lambda_792
tff(fact_10998_ATP_Olambda__793,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_real_bool,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun6526404207338086040t_bool(aa_set4447418679231215587t_bool(aTP_Lamm_ug(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uua))
        & pp(aa_real_bool(Uub,aa_nat_real(Uu,Uuc))) ) ) ).

% ATP.lambda_793
tff(fact_10999_ATP_Olambda__794,axiom,
    ! [Uu: fun_int_VEBT_VEBT,Uua: set_int,Uub: fun_VEBT_VEBT_bool,Uuc: int] :
      ( pp(aa_int_bool(aa_fun5862751214176358880t_bool(aa_set4588796106636019031t_bool(aTP_Lamm_ui(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uua))
        & pp(aa_VEBT_VEBT_bool(Uub,aa_int_VEBT_VEBT(Uu,Uuc))) ) ) ).

% ATP.lambda_794
tff(fact_11000_ATP_Olambda__795,axiom,
    ! [Uu: fun_int_real,Uua: set_int,Uub: fun_real_bool,Uuc: int] :
      ( pp(aa_int_bool(aa_fun3402113240869461820t_bool(aa_set559342874640923363t_bool(aTP_Lamm_uk(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uua))
        & pp(aa_real_bool(Uub,aa_int_real(Uu,Uuc))) ) ) ).

% ATP.lambda_795
tff(fact_11001_ATP_Olambda__796,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_dv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_796
tff(fact_11002_ATP_Olambda__797,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun1017860439744518025T_bool(aa_fun5042198764111188021T_bool(aTP_Lamm_eh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( 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_797
tff(fact_11003_ATP_Olambda__798,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3960896333048504454r_real,Uub: fun_Co3960896333048504454r_real,Uuc: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun7028086442024493327r_bool(aa_fun7037952363265197026r_bool(aTP_Lamm_ef(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(Uua,Uuc)),aa_Code_integer_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_798
tff(fact_11004_ATP_Olambda__799,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_ed(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_799
tff(fact_11005_ATP_Olambda__800,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_dx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_800
tff(fact_11006_ATP_Olambda__801,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_ej(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( 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_801
tff(fact_11007_ATP_Olambda__802,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_dz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_802
tff(fact_11008_ATP_Olambda__803,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_el(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( 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_803
tff(fact_11009_ATP_Olambda__804,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_eb(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_804
tff(fact_11010_ATP_Olambda__805,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( pp(aa_int_bool(aa_fun258058769140594373t_bool(aa_fun465274253410646749t_bool(aTP_Lamm_en(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( 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_805
tff(fact_11011_ATP_Olambda__806,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_VE5569070535936029429omplex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_eq(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_VEBT_VEBT_complex(Uua,Uuc)),aa_VEBT_VEBT_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_806
tff(fact_11012_ATP_Olambda__807,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_fc(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set8260762808118238015T_bool(member_VEBT_VEBT(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_807
tff(fact_11013_ATP_Olambda__808,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1900716531650368264omplex,Uub: fun_Co1900716531650368264omplex,Uuc: code_integer] :
      ( pp(aa_Code_integer_bool(aa_fun5910925830913328781r_bool(aa_fun6901856055265004766r_bool(aTP_Lamm_fa(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_Cod5621350732091595593omplex(Uua,Uuc)),aa_Cod5621350732091595593omplex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_808
tff(fact_11014_ATP_Olambda__809,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_ey(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_809
tff(fact_11015_ATP_Olambda__810,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_es(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_810
tff(fact_11016_ATP_Olambda__811,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_fe(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_811
tff(fact_11017_ATP_Olambda__812,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_eu(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_812
tff(fact_11018_ATP_Olambda__813,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_fg(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_813
tff(fact_11019_ATP_Olambda__814,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_ew(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_814
tff(fact_11020_ATP_Olambda__815,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_fi(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_815
tff(fact_11021_ATP_Olambda__816,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: bool] : aa_bool_assn(aa_nat_fun_bool_assn(aa_VEB5211276637198020136l_assn(aTP_Lamm_au(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,Uuc),aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Uu),Uub)))) ).

% ATP.lambda_816
tff(fact_11022_ATP_Olambda__817,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBTi,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_ow(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(Uub,Uua)),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(aa_VEB8608361877424424648i_bool(fequal_VEBT_VEBTi,Uuc),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uua),Uu)))) ).

% ATP.lambda_817
tff(fact_11023_ATP_Olambda__818,axiom,
    ! [Uu: nat,Uua: list_option_nat,Uub: array_option_nat,Uuc: option_nat] : aa_option_nat_assn(aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_ou(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_option_nat(Uub,Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),aa_nat_option_nat(nth_option_nat(Uua),Uu)))) ).

% ATP.lambda_818
tff(fact_11024_ATP_Olambda__819,axiom,
    ! [Uu: nat,Uua: list_bool,Uub: array_bool,Uuc: bool] : aa_bool_assn(aa_arr3483125744708554596l_assn(aa_lis2577335236791371354l_assn(aTP_Lamm_ov(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_bool(Uub,Uua)),aa_bool_assn(pure_assn,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,Uuc),aa_nat_bool(nth_bool(Uua),Uu)))) ).

% ATP.lambda_819
tff(fact_11025_ATP_Olambda__820,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: array_nat,Uuc: nat] : aa_nat_assn(aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_ox(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_nat(Uub,Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),aa_nat_nat(nth_nat(Uua),Uu)))) ).

% ATP.lambda_820
tff(fact_11026_ATP_Olambda__821,axiom,
    ! [Uu: nat,Uua: list_int,Uub: array_int,Uuc: int] : aa_int_assn(aa_arr1589534196239669886t_assn(aa_lis459371148865260371t_assn(aTP_Lamm_ot(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_int(Uub,Uua)),aa_bool_assn(pure_assn,aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),aa_nat_int(nth_int(Uua),Uu)))) ).

% ATP.lambda_821
tff(fact_11027_ATP_Olambda__822,axiom,
    ! [Uu: fun_VEBT_VEBTi_rat,Uua: rat,Uub: list_VEBT_VEBTi,Uuc: nat] : aa_nat_rat(aa_lis5007409733734533800at_rat(aa_rat8420674886958173181at_rat(aTP_Lamm_act(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_VEBT_VEBTi_rat(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_822
tff(fact_11028_ATP_Olambda__823,axiom,
    ! [Uu: fun_VEBT_VEBTi_int,Uua: int,Uub: list_VEBT_VEBTi,Uuc: nat] : aa_nat_int(aa_lis4337787272618733820at_int(aa_int7846649280651917989at_int(aTP_Lamm_acz(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_VEBT_VEBTi_int(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uub),Uuc))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_823
tff(fact_11029_ATP_Olambda__824,axiom,
    ! [Uu: fun_VEBT_VEBT_rat,Uua: rat,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_rat(aa_lis4303274161602456867at_rat(aa_rat7802996917357006812at_rat(aTP_Lamm_acu(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_VEBT_VEBT_rat(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_824
tff(fact_11030_ATP_Olambda__825,axiom,
    ! [Uu: fun_VEBT_VEBT_int,Uua: int,Uub: list_VEBT_VEBT,Uuc: nat] : aa_nat_int(aa_lis3633651700486656887at_int(aa_int4104624148011801308at_int(aTP_Lamm_ada(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_VEBT_VEBT_int(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uub),Uuc))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_825
tff(fact_11031_ATP_Olambda__826,axiom,
    ! [Uu: fun_real_rat,Uua: rat,Uub: list_real,Uuc: nat] : aa_nat_rat(aa_lis2767325350024641823at_rat(aa_rat1858428167516265844at_rat(aTP_Lamm_acv(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uu,aa_nat_real(nth_real(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_826
tff(fact_11032_ATP_Olambda__827,axiom,
    ! [Uu: fun_real_int,Uua: int,Uub: list_real,Uuc: nat] : aa_nat_int(aa_lis2097702888908841843at_int(aa_int464142701096502812at_int(aTP_Lamm_adb(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uu,aa_nat_real(nth_real(Uub),Uuc))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_827
tff(fact_11033_ATP_Olambda__828,axiom,
    ! [Uu: fun_bool_rat,Uua: rat,Uub: list_bool,Uuc: nat] : aa_nat_rat(aa_lis1392928297027365766at_rat(aa_rat954805563275384383at_rat(aTP_Lamm_acw(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_bool_rat(Uu,aa_nat_bool(nth_bool(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_828
tff(fact_11034_ATP_Olambda__829,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: list_nat,Uuc: nat] : aa_nat_rat(aa_lis7014846069874721091at_rat(aa_rat4396832037704353944at_rat(aTP_Lamm_acx(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_829
tff(fact_11035_ATP_Olambda__830,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: list_nat,Uuc: nat] : aa_nat_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_adc(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_830
tff(fact_11036_ATP_Olambda__831,axiom,
    ! [Uu: fun_int_rat,Uua: rat,Uub: list_int,Uuc: nat] : aa_nat_rat(aa_lis1742918661435654559at_rat(aa_rat8348276666120063220at_rat(aTP_Lamm_acy(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_831
tff(fact_11037_ATP_Olambda__832,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real,Uuc: real] : aa_real_real(aa_rea7340605660300375871l_real(aa_rea4187809805863855336l_real(aTP_Lamm_pr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uuc)) ).

% ATP.lambda_832
tff(fact_11038_ATP_Olambda__833,axiom,
    ! [Uu: fun_bool_int,Uua: int,Uub: bool,Uuc: int] : aa_int_int(aa_bool_fun_int_int(aa_int5463180763086053397nt_int(aTP_Lamm_pt(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(Uu,Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc)) ).

% ATP.lambda_833
tff(fact_11039_ATP_Olambda__834,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ps(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc)) ).

% ATP.lambda_834
tff(fact_11040_ATP_Olambda__835,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_aig(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_835
tff(fact_11041_ATP_Olambda__836,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_afg(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_836
tff(fact_11042_ATP_Olambda__837,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_afi(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_837
tff(fact_11043_ATP_Olambda__838,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_afe(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_838
tff(fact_11044_ATP_Olambda__839,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_afk(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_839
tff(fact_11045_ATP_Olambda__840,axiom,
    ! [Uu: set_complex,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co7124838579184476414x_bool,Uuc: code_integer] : aa_Cod5621350732091595593omplex(aa_fun7983975303565649482omplex(aa_fun843835629537646824omplex(aTP_Lamm_ahq(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_Cod3397930797123873268omplex(Uua,Uuc),collect_complex(aa_Cod6819470024986797159x_bool(aa_fun3619973020723243580x_bool(aTP_Lamm_ahp(Uu),Uub),Uuc))) ).

% ATP.lambda_840
tff(fact_11046_ATP_Olambda__841,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5750378413879446255x_bool,Uuc: int] : aa_int_complex(aa_fun751309451324587018omplex(aa_fun3438240386181387033omplex(aTP_Lamm_ahl(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int5354780257279585880x_bool(aa_fun6456964791495427196x_bool(aTP_Lamm_ahk(Uu),Uub),Uuc))) ).

% ATP.lambda_841
tff(fact_11047_ATP_Olambda__842,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na5941084329111710283x_bool,Uuc: nat] : aa_nat_complex(aa_fun2700696402887535882omplex(aa_fun8019143150332516725omplex(aTP_Lamm_ahg(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat5545486172511849908x_bool(aa_fun2551091230351774588x_bool(aTP_Lamm_ahf(Uu),Uub),Uuc))) ).

% ATP.lambda_842
tff(fact_11048_ATP_Olambda__843,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re3010880487650781423x_bool,Uuc: real] : aa_real_complex(aa_fun4873726854752153866omplex(aa_fun883667864959905433omplex(aTP_Lamm_ahc(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea5887071448072802392x_bool(aa_fun9179849694055305084x_bool(aTP_Lamm_ahb(Uu),Uub),Uuc))) ).

% ATP.lambda_843
tff(fact_11049_ATP_Olambda__844,axiom,
    ! [Uu: set_complex,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE2128610149277289041x_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6203732500129703078omplex(aa_fun7613540521737166425omplex(aTP_Lamm_agy(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_VEB5608662635949521omplex(Uua,Uuc),collect_complex(aa_VEB8450443329232412106x_bool(aa_fun5298830385910147808x_bool(aTP_Lamm_agx(Uu),Uub),Uuc))) ).

% ATP.lambda_844
tff(fact_11050_ATP_Olambda__845,axiom,
    ! [Uu: set_nat,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co8657364630979331932t_bool,Uuc: code_integer] : aa_Code_integer_nat(aa_fun2949368542467568330er_nat(aa_fun8366171557965857828er_nat(aTP_Lamm_agt(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Cod8670293052248272184at_nat(Uua,Uuc)),collect_nat(aa_Cod2498268374540730821t_bool(aa_fun498830667440607164t_bool(aTP_Lamm_ags(Uu),Uub),Uuc))) ).

% ATP.lambda_845
tff(fact_11051_ATP_Olambda__846,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co2841405903858895691t_bool,Uuc: complex] : aa_complex_nat(aa_fun9050545535516344074ex_nat(aa_fun2757644259618170323ex_nat(aTP_Lamm_ago(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_com5335841084339320244t_bool(aa_fun1991907153922069372t_bool(aTP_Lamm_agn(Uu),Uub),Uuc))) ).

% ATP.lambda_846
tff(fact_11052_ATP_Olambda__847,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_bool,Uuc: int] : aa_int_nat(aa_fun5132377469993263754nt_nat(aa_fun348247319442030677nt_nat(aTP_Lamm_agj(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_bool(aa_fun3602880909931171836t_bool(aTP_Lamm_agi(Uu),Uub),Uuc))) ).

% ATP.lambda_847
tff(fact_11053_ATP_Olambda__848,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_re9121486324101951821t_bool,Uuc: real] : aa_real_nat(aa_fun5804151645385689226al_nat(aa_fun8436216498351771861al_nat(aTP_Lamm_agf(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_bool(aa_fun5702564393759107580t_bool(aTP_Lamm_age(Uu),Uub),Uuc))) ).

% ATP.lambda_848
tff(fact_11054_ATP_Olambda__849,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE3763956218935105199t_bool,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun1096289392377091562BT_nat(aa_fun7476968357888788569BT_nat(aTP_Lamm_agb(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc)),collect_nat(aa_VEB4832260192069861672t_bool(aa_fun5431641998072558748t_bool(aTP_Lamm_aga(Uu),Uub),Uuc))) ).

% ATP.lambda_849
tff(fact_11055_ATP_Olambda__850,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_gh(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),aa_VEB7679222860772546456on_nat(vEBT_vebt_mint,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uu))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_850
tff(fact_11056_ATP_Olambda__851,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_gg(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),aa_VEB7679222860772546456on_nat(vEBT_vebt_maxt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uu))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_851
tff(fact_11057_ATP_Olambda__852,axiom,
    ! [Uu: assn,Uua: fun_VE905575822219544865i_assn,Uub: vEBT_VEBT,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(aa_fun4659777589554175228i_assn(aTP_Lamm_ia(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_852
tff(fact_11058_ATP_Olambda__853,axiom,
    ! [Uu: assn,Uua: fun_VE6334927249922962060t_assn,Uub: vEBT_VEBT,Uuc: option_nat] : aa_option_nat_assn(aa_VEB931861714549014589t_assn(aa_fun1240582285364316732t_assn(aTP_Lamm_hq(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_option_nat_assn(aa_VEB931861714549014589t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_853
tff(fact_11059_ATP_Olambda__854,axiom,
    ! [Uu: assn,Uua: fun_VE1323945509527881313l_assn,Uub: vEBT_VEBT,Uuc: bool] : aa_bool_assn(aa_VEB2879522196611654106l_assn(aa_fun6755658226996116224l_assn(aTP_Lamm_hu(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_bool_assn(aa_VEB2879522196611654106l_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_854
tff(fact_11060_ATP_Olambda__855,axiom,
    ! [Uu: assn,Uua: fun_VE4931415397359990460t_assn,Uub: vEBT_VEBT,Uuc: nat] : aa_nat_assn(aa_VEB8640158415945897581t_assn(aa_fun7634423771582967228t_assn(aTP_Lamm_ic(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_nat_assn(aa_VEB8640158415945897581t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_855
tff(fact_11061_ATP_Olambda__856,axiom,
    ! [Uu: assn,Uua: fun_bo7535889243153246212i_assn,Uub: bool,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(aa_fun3930196489193831100i_assn(aTP_Lamm_hy(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_856
tff(fact_11062_ATP_Olambda__857,axiom,
    ! [Uu: assn,Uua: fun_bo1795969981296709615t_assn,Uub: bool,Uuc: option_nat] : aa_option_nat_assn(aa_boo2787173234663865760t_assn(aa_fun9080105975142006140t_assn(aTP_Lamm_hs(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_option_nat_assn(aa_boo2787173234663865760t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_857
tff(fact_11063_ATP_Olambda__858,axiom,
    ! [Uu: assn,Uua: fun_bo8172996160782635326l_assn,Uub: bool,Uuc: bool] : aa_bool_assn(aa_boo2938360491098410167l_assn(aa_fun6282432130954631226l_assn(aTP_Lamm_hw(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_bool_assn(aa_boo2938360491098410167l_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_858
tff(fact_11064_ATP_Olambda__859,axiom,
    ! [Uu: assn,Uua: fun_bo8438563643002758303t_assn,Uub: bool,Uuc: nat] : aa_nat_assn(aa_bool_fun_nat_assn(aa_fun8165606209698844796t_assn(aTP_Lamm_ie(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_nat_assn(aa_bool_fun_nat_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_859
tff(fact_11065_ATP_Olambda__860,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_alb(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_860
tff(fact_11066_ATP_Olambda__861,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_akz(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(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,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_861
tff(fact_11067_ATP_Olambda__862,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ald(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_862
tff(fact_11068_ATP_Olambda__863,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_alf(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_863
tff(fact_11069_ATP_Olambda__864,axiom,
    ! [Uu: fun_VE2561185598974572172i_assn,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBTi,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9992151435860550l_assn(aa_lis4074100605738197896l_assn(aTP_Lamm_oj(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1891944875198410415_VEBTi(Uu,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_864
tff(fact_11070_ATP_Olambda__865,axiom,
    ! [Uu: fun_VE5577048379444288083T_assn,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBT,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis183111350483865165l_assn(aTP_Lamm_ok(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7265847600308530106T_VEBT(Uu,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_865
tff(fact_11071_ATP_Olambda__866,axiom,
    ! [Uu: fun_VE1225457145571040039t_assn,Uua: list_VEBT_VEBTi,Uub: list_nat,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis3503319847846646691l_assn(aTP_Lamm_oi(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8930081998596925642Ti_nat(Uu,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_866
tff(fact_11072_ATP_Olambda__867,axiom,
    ! [Uu: fun_VE5641251438918192003t_assn,Uua: list_VEBT_VEBTi,Uub: list_int,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis7389664606029993405l_assn(aa_lis1781686709860263423l_assn(aTP_Lamm_ol(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8927591528087875366Ti_int(Uu,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_867
tff(fact_11073_ATP_Olambda__868,axiom,
    ! [Uu: fun_VE1571388752190420990T_assn,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis7769146593908314002l_assn(aTP_Lamm_on(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1279224858307276611T_VEBT(Uu,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_868
tff(fact_11074_ATP_Olambda__869,axiom,
    ! [Uu: fun_VE4931415397359990460t_assn,Uua: list_VEBT_VEBT,Uub: list_nat,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis5809874432478338334l_assn(aTP_Lamm_om(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(Uu,Uua),Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_869
tff(fact_11075_ATP_Olambda__870,axiom,
    ! [Uu: fun_VE123837653852366616t_assn,Uua: list_VEBT_VEBT,Uub: list_int,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis7389664606029993405l_assn(aa_lis4088241294491955066l_assn(aTP_Lamm_oo(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8294436054247626077BT_int(Uu,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_870
tff(fact_11076_ATP_Olambda__871,axiom,
    ! [Uu: fun_re560840401387337987i_assn,Uua: list_real,Uub: list_VEBT_VEBTi,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9992151435860550l_assn(aa_lis3027762689479947775l_assn(aTP_Lamm_oq(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L9060850011106065574_VEBTi(Uu,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_871
tff(fact_11077_ATP_Olambda__872,axiom,
    ! [Uu: fun_re8050108611700907676T_assn,Uua: list_real,Uub: list_VEBT_VEBT,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis7705788952316199574l_assn(aTP_Lamm_or(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L4595930785310033027T_VEBT(Uu,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_872
tff(fact_11078_ATP_Olambda__873,axiom,
    ! [Uu: fun_re574865750389657758t_assn,Uua: list_real,Uub: list_nat,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis4366942040065530138l_assn(aTP_Lamm_op(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1446010312343316929al_nat(Uu,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_873
tff(fact_11079_ATP_Olambda__874,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBTi,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: array_VEBT_VEBTi] : aa_arr8637723718858317150i_assn(aa_VEB7841561841333234298i_assn(aa_arr6331738488200487899i_assn(aa_lis1967528460011584869i_assn(aTP_Lamm_pc(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(Uub,list_u6098035379799741383_VEBTi(Uua,Uu,Uuc))),aa_bool_assn(pure_assn,fequal8089995073349378887_VEBTi(Uud,Uub))) ).

% ATP.lambda_874
tff(fact_11080_ATP_Olambda__875,axiom,
    ! [Uu: nat,Uua: list_bool,Uub: array_bool,Uuc: bool,Uud: array_bool] : aa_array_bool_assn(aa_boo8717665131543412656l_assn(aa_arr1850246612958046993l_assn(aa_lis1773218934604694151l_assn(aTP_Lamm_oz(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_bool(Uub,list_update_bool(Uua,Uu,Uuc))),aa_bool_assn(pure_assn,fequal_array_bool(Uud,Uub))) ).

% ATP.lambda_875
tff(fact_11081_ATP_Olambda__876,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: array_nat,Uuc: nat,Uud: array_nat] : aa_array_nat_assn(aa_nat8392338049721622586t_assn(aa_arr4802002299793635008t_assn(aa_lis5007279772662053221t_assn(aTP_Lamm_pa(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_nat(Uub,list_update_nat(Uua,Uu,Uuc))),aa_bool_assn(pure_assn,fequal_array_nat(Uud,Uub))) ).

% ATP.lambda_876
tff(fact_11082_ATP_Olambda__877,axiom,
    ! [Uu: nat,Uua: list_int,Uub: array_int,Uuc: int,Uud: array_int] : aa_array_int_assn(aa_int2412489125148462906t_assn(aa_arr8066481932369824868t_assn(aa_lis2681267339321482597t_assn(aTP_Lamm_pb(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_int(Uub,list_update_int(Uua,Uu,Uuc))),aa_bool_assn(pure_assn,fequal_array_int(Uud,Uub))) ).

% ATP.lambda_877
tff(fact_11083_ATP_Olambda__878,axiom,
    ! [Uu: assn,Uua: fun_VE905575822219544865i_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_nat6700791848136694008i_assn(aa_VEB320677594755043119i_assn(aa_fun8635824850342594777i_assn(aTP_Lamm_ib(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(Uua,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(Uuc),Uub)),Uud)) ).

% ATP.lambda_878
tff(fact_11084_ATP_Olambda__879,axiom,
    ! [Uu: assn,Uua: fun_VE6334927249922962060t_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_option_nat] : aa_lis1398067190886521878t_assn(aa_nat5248546159757461219t_assn(aa_VEB2926451386496479642t_assn(aa_fun7129419364150962969t_assn(aTP_Lamm_hr(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_lis1398067190886521878t_assn(vEBT_L8010285020845282001on_nat(Uua,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(Uuc),Uub)),Uud)) ).

% ATP.lambda_879
tff(fact_11085_ATP_Olambda__880,axiom,
    ! [Uu: assn,Uua: fun_VE1323945509527881313l_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_bool] : aa_list_bool_assn(aa_nat2784677144853144906l_assn(aa_VEB3184745993655927913l_assn(aa_fun7303525306808986255l_assn(aTP_Lamm_hv(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_list_bool_assn(vEBT_L5320321625355440422T_bool(Uua,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(Uuc),Uub)),Uud)) ).

% ATP.lambda_880
tff(fact_11086_ATP_Olambda__881,axiom,
    ! [Uu: assn,Uua: fun_VE4931415397359990460t_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_nat] : aa_list_nat_assn(aa_nat7967224596217534867t_assn(aa_VEB8402767592425156170t_assn(aa_fun6018615163312975129t_assn(aTP_Lamm_id(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(Uua,aa_VEB5888937730619358530T_VEBT(replicate_VEBT_VEBT(Uuc),Uub)),Uud)) ).

% ATP.lambda_881
tff(fact_11087_ATP_Olambda__882,axiom,
    ! [Uu: assn,Uua: fun_bo7535889243153246212i_assn,Uub: bool,Uuc: nat,Uud: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_nat6700791848136694008i_assn(aa_boo3536769854965821074i_assn(aa_fun2584265494025193113i_assn(aTP_Lamm_hz(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_lis2508312872377092907i_assn(vEBT_L1689158095293668553_VEBTi(Uua,replicate_bool(Uuc,Uub)),Uud)) ).

% ATP.lambda_882
tff(fact_11088_ATP_Olambda__883,axiom,
    ! [Uu: assn,Uua: fun_bo1795969981296709615t_assn,Uub: bool,Uuc: nat,Uud: list_option_nat] : aa_lis1398067190886521878t_assn(aa_nat5248546159757461219t_assn(aa_boo6521129556348374909t_assn(aa_fun7381918676215250521t_assn(aTP_Lamm_ht(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_lis1398067190886521878t_assn(vEBT_L5912725412549479476on_nat(Uua,replicate_bool(Uuc,Uub)),Uud)) ).

% ATP.lambda_883
tff(fact_11089_ATP_Olambda__884,axiom,
    ! [Uu: assn,Uua: fun_bo8172996160782635326l_assn,Uub: bool,Uuc: nat,Uud: list_bool] : aa_list_bool_assn(aa_nat2784677144853144906l_assn(aa_boo5646914333422710470l_assn(aa_fun2051762549316634313l_assn(aTP_Lamm_hx(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_list_bool_assn(vEBT_L6562251693209944067l_bool(Uua,replicate_bool(Uuc,Uub)),Uud)) ).

% ATP.lambda_884
tff(fact_11090_ATP_Olambda__885,axiom,
    ! [Uu: assn,Uua: fun_bo8438563643002758303t_assn,Uub: bool,Uuc: nat,Uud: list_nat] : aa_list_nat_assn(aa_nat7967224596217534867t_assn(aa_boo7576344649022168237t_assn(aa_fun7875716396929150937t_assn(aTP_Lamm_if(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_list_nat_assn(vEBT_L7520539408223738084ol_nat(Uua,replicate_bool(Uuc,Uub)),Uud)) ).

% ATP.lambda_885
tff(fact_11091_ATP_Olambda__886,axiom,
    ! [Uu: fun_VE2561185598974572172i_assn,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBTi,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis5217563234146327453n_assn(aa_lis8873771691224567519n_assn(aTP_Lamm_pd(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uua),Uuc)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uub),Uuc))) ).

% ATP.lambda_886
tff(fact_11092_ATP_Olambda__887,axiom,
    ! [Uu: fun_VE5577048379444288083T_assn,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBT,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis2699940410532524056n_assn(aa_lis8759289953362995748n_assn(aTP_Lamm_pe(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uua),Uuc)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uub),Uuc))) ).

% ATP.lambda_887
tff(fact_11093_ATP_Olambda__888,axiom,
    ! [Uu: fun_VE1225457145571040039t_assn,Uua: list_VEBT_VEBTi,Uub: list_nat,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis8171252396799902776n_assn(aa_lis2805659845025431162n_assn(aTP_Lamm_qe(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_nat_assn(aa_VEB8068455526268438248t_assn(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uua),Uuc)),aa_nat_nat(nth_nat(Uub),Uuc))) ).

% ATP.lambda_888
tff(fact_11094_ATP_Olambda__889,axiom,
    ! [Uu: fun_VE5641251438918192003t_assn,Uua: list_VEBT_VEBTi,Uub: list_int,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis157064186919824532n_assn(aa_lis4014843672000128726n_assn(aTP_Lamm_qi(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_int_assn(aa_VEB9043990991474615620t_assn(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uua),Uuc)),aa_nat_int(nth_int(Uub),Uuc))) ).

% ATP.lambda_889
tff(fact_11095_ATP_Olambda__890,axiom,
    ! [Uu: fun_VE905575822219544865i_assn,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis5217563234146327453n_assn(aa_lis4092487795791266010n_assn(aTP_Lamm_qb(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uuc)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uub),Uuc))) ).

% ATP.lambda_890
tff(fact_11096_ATP_Olambda__891,axiom,
    ! [Uu: fun_VE1571388752190420990T_assn,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis2699940410532524056n_assn(aa_lis4214038909079116777n_assn(aTP_Lamm_pf(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uuc)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uub),Uuc))) ).

% ATP.lambda_891
tff(fact_11097_ATP_Olambda__892,axiom,
    ! [Uu: fun_VE4931415397359990460t_assn,Uua: list_VEBT_VEBT,Uub: list_nat,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis8171252396799902776n_assn(aa_lis7929737622710957941n_assn(aTP_Lamm_qg(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_nat_assn(aa_VEB8640158415945897581t_assn(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uuc)),aa_nat_nat(nth_nat(Uub),Uuc))) ).

% ATP.lambda_892
tff(fact_11098_ATP_Olambda__893,axiom,
    ! [Uu: fun_VE123837653852366616t_assn,Uua: list_VEBT_VEBT,Uub: list_int,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis157064186919824532n_assn(aa_lis9138921449685655505n_assn(aTP_Lamm_qk(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_int_assn(aa_VEB392321844297299145t_assn(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uuc)),aa_nat_int(nth_int(Uub),Uuc))) ).

% ATP.lambda_893
tff(fact_11099_ATP_Olambda__894,axiom,
    ! [Uu: fun_re560840401387337987i_assn,Uua: list_real,Uub: list_VEBT_VEBTi,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis5217563234146327453n_assn(aa_lis6806030848160413398n_assn(aTP_Lamm_pg(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(Uu,aa_nat_real(nth_real(Uua),Uuc)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uub),Uuc))) ).

% ATP.lambda_894
tff(fact_11100_ATP_Olambda__895,axiom,
    ! [Uu: fun_re8050108611700907676T_assn,Uua: list_real,Uub: list_VEBT_VEBT,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis2699940410532524056n_assn(aa_lis7292822999309854957n_assn(aTP_Lamm_ph(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(Uu,aa_nat_real(nth_real(Uua),Uuc)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uub),Uuc))) ).

% ATP.lambda_895
tff(fact_11101_ATP_Olambda__896,axiom,
    ! [Uu: fun_bo7535889243153246212i_assn,Uua: list_bool,Uub: list_VEBT_VEBTi,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis5217563234146327453n_assn(aa_lis2049715178687361725n_assn(aTP_Lamm_pi(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(Uu,aa_nat_bool(nth_bool(Uua),Uuc)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uub),Uuc))) ).

% ATP.lambda_896
tff(fact_11102_ATP_Olambda__897,axiom,
    ! [Uu: fun_bo5394993412183903963T_assn,Uua: list_bool,Uub: list_VEBT_VEBT,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis2699940410532524056n_assn(aa_lis3096371939905232838n_assn(aTP_Lamm_pj(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(Uu,aa_nat_bool(nth_bool(Uua),Uuc)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uub),Uuc))) ).

% ATP.lambda_897
tff(fact_11103_ATP_Olambda__898,axiom,
    ! [Uu: fun_na6467670216208870567i_assn,Uua: list_nat,Uub: list_VEBT_VEBTi,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis5217563234146327453n_assn(aa_lis1134338210885690618n_assn(aTP_Lamm_pk(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(Uu,aa_nat_nat(nth_nat(Uua),Uuc)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uub),Uuc))) ).

% ATP.lambda_898
tff(fact_11104_ATP_Olambda__899,axiom,
    ! [Uu: fun_na4489259242115493880T_assn,Uua: list_nat,Uub: list_VEBT_VEBT,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis2699940410532524056n_assn(aa_lis3837078307170607049n_assn(aTP_Lamm_pl(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(Uu,aa_nat_nat(nth_nat(Uua),Uuc)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uub),Uuc))) ).

% ATP.lambda_899
tff(fact_11105_ATP_Olambda__900,axiom,
    ! [Uu: fun_in1903134756389476099i_assn,Uua: list_int,Uub: list_VEBT_VEBTi,Uuc: nat,Uud: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis5217563234146327453n_assn(aa_lis1861013024967981910n_assn(aTP_Lamm_pm(Uu),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uud),aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(Uu,aa_nat_int(nth_int(Uua),Uuc)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Uub),Uuc))) ).

% ATP.lambda_900
tff(fact_11106_ATP_Olambda__901,axiom,
    ! [Uu: fun_VE2561185598974572172i_assn,Uua: vEBT_VEBTi,Uub: list_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBTi,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis5217563234146327453n_assn(aa_VEB7059816536500214223n_assn(aa_lis4146167562466521361n_assn(aa_VEB5563403434209864643n_assn(aTP_Lamm_py(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Uua,Uub)),Uue)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Uuc,Uud)),Uue))) ).

% ATP.lambda_901
tff(fact_11107_ATP_Olambda__902,axiom,
    ! [Uu: fun_VE5577048379444288083T_assn,Uua: vEBT_VEBTi,Uub: list_VEBT_VEBTi,Uuc: vEBT_VEBT,Uud: list_VEBT_VEBT,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis2699940410532524056n_assn(aa_VEB3810757911076339055n_assn(aa_lis1293208971169636985n_assn(aa_VEB7709549983103476651n_assn(aTP_Lamm_pz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Uua,Uub)),Uue)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Uuc,Uud)),Uue))) ).

% ATP.lambda_902
tff(fact_11108_ATP_Olambda__903,axiom,
    ! [Uu: fun_VE1225457145571040039t_assn,Uua: vEBT_VEBTi,Uub: list_VEBT_VEBTi,Uuc: nat,Uud: list_nat,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis8171252396799902776n_assn(aa_nat394924699201469189n_assn(aa_lis8390372872860913351n_assn(aa_VEB1074492319160947833n_assn(aTP_Lamm_qd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_nat_assn(aa_VEB8068455526268438248t_assn(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Uua,Uub)),Uue)),aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(Uuc),Uud)),Uue))) ).

% ATP.lambda_903
tff(fact_11109_ATP_Olambda__904,axiom,
    ! [Uu: fun_VE5641251438918192003t_assn,Uua: vEBT_VEBTi,Uub: list_VEBT_VEBTi,Uuc: int,Uud: list_int,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis157064186919824532n_assn(aa_int5700688002552083389n_assn(aa_lis8289800597781125503n_assn(aa_VEB219408230628803377n_assn(aTP_Lamm_qh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_int_assn(aa_VEB9043990991474615620t_assn(Uu,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Uua,Uub)),Uue)),aa_nat_int(nth_int(aa_list_int_list_int(cons_int(Uuc),Uud)),Uue))) ).

% ATP.lambda_904
tff(fact_11110_ATP_Olambda__905,axiom,
    ! [Uu: fun_VE905575822219544865i_assn,Uua: vEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBTi,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis5217563234146327453n_assn(aa_VEB7059816536500214223n_assn(aa_lis7703346581310142988n_assn(aa_VEB8713780084012957539n_assn(aTP_Lamm_qa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Uua,Uub)),Uue)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Uuc,Uud)),Uue))) ).

% ATP.lambda_905
tff(fact_11111_ATP_Olambda__906,axiom,
    ! [Uu: fun_VE1571388752190420990T_assn,Uua: vEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: vEBT_VEBT,Uud: list_VEBT_VEBT,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis2699940410532524056n_assn(aa_VEB3810757911076339055n_assn(aa_lis6098897284473852916n_assn(aa_VEB7190415443829630795n_assn(aTP_Lamm_qc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Uua,Uub)),Uue)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Uuc,Uud)),Uue))) ).

% ATP.lambda_906
tff(fact_11112_ATP_Olambda__907,axiom,
    ! [Uu: fun_VE4931415397359990460t_assn,Uua: vEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: nat,Uud: list_nat,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis8171252396799902776n_assn(aa_nat394924699201469189n_assn(aa_lis8752055264758851010n_assn(aa_VEB7545035127168460825n_assn(aTP_Lamm_qf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_nat_assn(aa_VEB8640158415945897581t_assn(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Uua,Uub)),Uue)),aa_nat_nat(nth_nat(aa_list_nat_list_nat(cons_nat(Uuc),Uud)),Uue))) ).

% ATP.lambda_907
tff(fact_11113_ATP_Olambda__908,axiom,
    ! [Uu: fun_VE123837653852366616t_assn,Uua: vEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: int,Uud: list_int,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis157064186919824532n_assn(aa_int5700688002552083389n_assn(aa_lis8651482989679063162n_assn(aa_VEB6689951038636316369n_assn(aTP_Lamm_qj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_int_assn(aa_VEB392321844297299145t_assn(Uu,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Uua,Uub)),Uue)),aa_nat_int(nth_int(aa_list_int_list_int(cons_int(Uuc),Uud)),Uue))) ).

% ATP.lambda_908
tff(fact_11114_ATP_Olambda__909,axiom,
    ! [Uu: fun_re560840401387337987i_assn,Uua: real,Uub: list_real,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBTi,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis5217563234146327453n_assn(aa_VEB7059816536500214223n_assn(aa_lis9048447917024140296n_assn(aa_rea7151089224645626673n_assn(aTP_Lamm_ql(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(Uu,aa_nat_real(nth_real(cons_real(Uua,Uub)),Uue)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Uuc,Uud)),Uue))) ).

% ATP.lambda_909
tff(fact_11115_ATP_Olambda__910,axiom,
    ! [Uu: fun_re8050108611700907676T_assn,Uua: real,Uub: list_real,Uuc: vEBT_VEBT,Uud: list_VEBT_VEBT,Uue: nat,Uuf: assn] : aa_assn_assn(aa_nat_fun_assn_assn(aa_lis2699940410532524056n_assn(aa_VEB3810757911076339055n_assn(aa_lis7268569422076391408n_assn(aa_rea588685846782134297n_assn(aTP_Lamm_qm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uuf),aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(Uu,aa_nat_real(nth_real(cons_real(Uua,Uub)),Uue)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Uuc,Uud)),Uue))) ).

% ATP.lambda_910
tff(fact_11116_ATP_Olambda__911,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_uv(Uu),Uua) = Uu ).

% ATP.lambda_911
tff(fact_11117_ATP_Olambda__912,axiom,
    ! [Uu: vEBT_VEBT,Uua: real] : aa_real_VEBT_VEBT(aTP_Lamm_ii(Uu),Uua) = Uu ).

% ATP.lambda_912
tff(fact_11118_ATP_Olambda__913,axiom,
    ! [Uu: vEBT_VEBT,Uua: bool] : aa_bool_VEBT_VEBT(aTP_Lamm_ik(Uu),Uua) = Uu ).

% ATP.lambda_913
tff(fact_11119_ATP_Olambda__914,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat_VEBT_VEBT(aTP_Lamm_in(Uu),Uua) = Uu ).

% ATP.lambda_914
tff(fact_11120_ATP_Olambda__915,axiom,
    ! [Uu: vEBT_VEBT,Uua: int] : aa_int_VEBT_VEBT(aTP_Lamm_ip(Uu),Uua) = Uu ).

% ATP.lambda_915
tff(fact_11121_ATP_Olambda__916,axiom,
    ! [Uu: real,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_ih(Uu),Uua) = Uu ).

% ATP.lambda_916
tff(fact_11122_ATP_Olambda__917,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_ux(Uu),Uua) = Uu ).

% ATP.lambda_917
tff(fact_11123_ATP_Olambda__918,axiom,
    ! [Uu: real,Uua: bool] : aa_bool_real(aTP_Lamm_vc(Uu),Uua) = Uu ).

% ATP.lambda_918
tff(fact_11124_ATP_Olambda__919,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_uw(Uu),Uua) = Uu ).

% ATP.lambda_919
tff(fact_11125_ATP_Olambda__920,axiom,
    ! [Uu: real,Uua: int] : aa_int_real(aTP_Lamm_uy(Uu),Uua) = Uu ).

% ATP.lambda_920
tff(fact_11126_ATP_Olambda__921,axiom,
    ! [Uu: bool,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_uz(Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_921
tff(fact_11127_ATP_Olambda__922,axiom,
    ! [Uu: bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ij(Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_922
tff(fact_11128_ATP_Olambda__923,axiom,
    ! [Uu: bool,Uua: bool] :
      ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(aTP_Lamm_il,Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_923
tff(fact_11129_ATP_Olambda__924,axiom,
    ! [Uu: bool,Uua: nat] :
      ( pp(aa_nat_bool(aa_bool_fun_nat_bool(aTP_Lamm_io,Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_924
tff(fact_11130_ATP_Olambda__925,axiom,
    ! [Uu: set_real,Uua: real] : aa_real_set_real(aTP_Lamm_zv(Uu),Uua) = Uu ).

% ATP.lambda_925
tff(fact_11131_ATP_Olambda__926,axiom,
    ! [Uu: set_real,Uua: bool] : aa_bool_set_real(aTP_Lamm_zy(Uu),Uua) = Uu ).

% ATP.lambda_926
tff(fact_11132_ATP_Olambda__927,axiom,
    ! [Uu: set_real,Uua: nat] : aa_nat_set_real(aTP_Lamm_aab(Uu),Uua) = Uu ).

% ATP.lambda_927
tff(fact_11133_ATP_Olambda__928,axiom,
    ! [Uu: set_real,Uua: int] : aa_int_set_real(aTP_Lamm_aae(Uu),Uua) = Uu ).

% ATP.lambda_928
tff(fact_11134_ATP_Olambda__929,axiom,
    ! [Uu: set_bool,Uua: real] : aa_real_set_bool(aTP_Lamm_zw(Uu),Uua) = Uu ).

% ATP.lambda_929
tff(fact_11135_ATP_Olambda__930,axiom,
    ! [Uu: set_bool,Uua: bool] : aa_bool_set_bool(aTP_Lamm_zz(Uu),Uua) = Uu ).

% ATP.lambda_930
tff(fact_11136_ATP_Olambda__931,axiom,
    ! [Uu: set_bool,Uua: nat] : aa_nat_set_bool(aTP_Lamm_aac(Uu),Uua) = Uu ).

% ATP.lambda_931
tff(fact_11137_ATP_Olambda__932,axiom,
    ! [Uu: set_nat,Uua: real] : aa_real_set_nat(aTP_Lamm_zr(Uu),Uua) = Uu ).

% ATP.lambda_932
tff(fact_11138_ATP_Olambda__933,axiom,
    ! [Uu: set_nat,Uua: bool] : aa_bool_set_nat(aTP_Lamm_zs(Uu),Uua) = Uu ).

% ATP.lambda_933
tff(fact_11139_ATP_Olambda__934,axiom,
    ! [Uu: set_nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_zt(Uu),Uua) = Uu ).

% ATP.lambda_934
tff(fact_11140_ATP_Olambda__935,axiom,
    ! [Uu: set_nat,Uua: int] : aa_int_set_nat(aTP_Lamm_zu(Uu),Uua) = Uu ).

% ATP.lambda_935
tff(fact_11141_ATP_Olambda__936,axiom,
    ! [Uu: set_int,Uua: real] : aa_real_set_int(aTP_Lamm_zx(Uu),Uua) = Uu ).

% ATP.lambda_936
tff(fact_11142_ATP_Olambda__937,axiom,
    ! [Uu: set_int,Uua: bool] : aa_bool_set_int(aTP_Lamm_aaa(Uu),Uua) = Uu ).

% ATP.lambda_937
tff(fact_11143_ATP_Olambda__938,axiom,
    ! [Uu: set_int,Uua: nat] : aa_nat_set_int(aTP_Lamm_aad(Uu),Uua) = Uu ).

% ATP.lambda_938
tff(fact_11144_ATP_Olambda__939,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_nat(aTP_Lamm_ig(Uu),Uua) = Uu ).

% ATP.lambda_939
tff(fact_11145_ATP_Olambda__940,axiom,
    ! [Uu: nat,Uua: code_integer] : aa_Code_integer_nat(aTP_Lamm_xi(Uu),Uua) = Uu ).

% ATP.lambda_940
tff(fact_11146_ATP_Olambda__941,axiom,
    ! [Uu: nat,Uua: complex] : aa_complex_nat(aTP_Lamm_xh(Uu),Uua) = Uu ).

% ATP.lambda_941
tff(fact_11147_ATP_Olambda__942,axiom,
    ! [Uu: nat,Uua: real] : aa_real_nat(aTP_Lamm_va(Uu),Uua) = Uu ).

% ATP.lambda_942
tff(fact_11148_ATP_Olambda__943,axiom,
    ! [Uu: nat,Uua: bool] : aa_bool_nat(aTP_Lamm_vd(Uu),Uua) = Uu ).

% ATP.lambda_943
tff(fact_11149_ATP_Olambda__944,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_im,Uu),Uua) = Uu ).

% ATP.lambda_944
tff(fact_11150_ATP_Olambda__945,axiom,
    ! [Uu: nat,Uua: int] : aa_int_nat(aTP_Lamm_xg(Uu),Uua) = Uu ).

% ATP.lambda_945
tff(fact_11151_ATP_Olambda__946,axiom,
    ! [Uu: int,Uua: code_integer] : aa_Code_integer_int(aTP_Lamm_xk(Uu),Uua) = Uu ).

% ATP.lambda_946
tff(fact_11152_ATP_Olambda__947,axiom,
    ! [Uu: int,Uua: complex] : aa_complex_int(aTP_Lamm_xj(Uu),Uua) = Uu ).

% ATP.lambda_947
tff(fact_11153_ATP_Olambda__948,axiom,
    ! [Uu: int,Uua: real] : aa_real_int(aTP_Lamm_vb(Uu),Uua) = Uu ).

% ATP.lambda_948
tff(fact_11154_ATP_Olambda__949,axiom,
    ! [Uu: int,Uua: bool] : aa_bool_int(aTP_Lamm_ve(Uu),Uua) = Uu ).

% ATP.lambda_949
tff(fact_11155_ATP_Olambda__950,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_xe(Uu),Uua) = Uu ).

% ATP.lambda_950
tff(fact_11156_ATP_Olambda__951,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_xf,Uu),Uua) = Uu ).

% ATP.lambda_951
tff(fact_11157_ATP_Olambda__952,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ha,Uu),Uua))
    <=> $true ) ).

% ATP.lambda_952
tff(fact_11158_ATP_Olambda__953,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_hb,Uu),Uua))
    <=> $true ) ).

% ATP.lambda_953
tff(fact_11159_ATP_Olambda__954,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_ss,Uu) = Uu ).

% ATP.lambda_954
tff(fact_11160_ATP_Olambda__955,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_do,Uu) = Uu ).

% ATP.lambda_955
tff(fact_11161_ATP_Olambda__956,axiom,
    ! [Uu: assn] : aa_assn_assn(aTP_Lamm_dn,Uu) = Uu ).

% ATP.lambda_956
tff(fact_11162_ATP_Olambda__957,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_dj,Uu) = Uu ).

% ATP.lambda_957
tff(fact_11163_ATP_Olambda__958,axiom,
    ! [Uu: bool] :
      ( pp(aa_bool_bool(aTP_Lamm_st,Uu))
    <=> pp(Uu) ) ).

% ATP.lambda_958
tff(fact_11164_ATP_Olambda__959,axiom,
    ! [Uu: set_nat] : aa_set_nat_set_nat(aTP_Lamm_zp,Uu) = Uu ).

% ATP.lambda_959
tff(fact_11165_ATP_Olambda__960,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_dk,Uu) = Uu ).

% ATP.lambda_960
tff(fact_11166_ATP_Olambda__961,axiom,
    ! [Uu: num] : aa_num_num(aTP_Lamm_su,Uu) = Uu ).

% ATP.lambda_961
tff(fact_11167_ATP_Olambda__962,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_dl,Uu) = Uu ).

% ATP.lambda_962
tff(fact_11168_ATP_Olambda__963,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_dm,Uu) = Uu ).

% ATP.lambda_963
tff(fact_11169_ATP_Olambda__964,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_zq,Uu) = bot_bot_set_nat ).

% ATP.lambda_964
tff(fact_11170_ATP_Olambda__965,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_am,Uu) = zero_zero_complex ).

% ATP.lambda_965
tff(fact_11171_ATP_Olambda__966,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ai,Uu) = zero_zero_real ).

% ATP.lambda_966
tff(fact_11172_ATP_Olambda__967,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_aj,Uu) = zero_zero_rat ).

% ATP.lambda_967
tff(fact_11173_ATP_Olambda__968,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_add,Uu) = zero_zero_real ).

% ATP.lambda_968
tff(fact_11174_ATP_Olambda__969,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ak,Uu) = zero_zero_nat ).

% ATP.lambda_969
tff(fact_11175_ATP_Olambda__970,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_al,Uu) = zero_zero_int ).

% ATP.lambda_970
tff(fact_11176_ATP_Olambda__971,axiom,
    ! [Uu: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ax,Uu))
    <=> $false ) ).

% ATP.lambda_971
tff(fact_11177_ATP_Olambda__972,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_ba,Uu))
    <=> $false ) ).

% ATP.lambda_972
tff(fact_11178_ATP_Olambda__973,axiom,
    ! [Uu: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ay,Uu))
    <=> $false ) ).

% ATP.lambda_973
tff(fact_11179_ATP_Olambda__974,axiom,
    ! [Uu: bool] :
      ( pp(aa_bool_bool(aTP_Lamm_bb,Uu))
    <=> $false ) ).

% ATP.lambda_974
tff(fact_11180_ATP_Olambda__975,axiom,
    ! [Uu: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_az,Uu))
    <=> $false ) ).

% ATP.lambda_975
tff(fact_11181_ATP_Olambda__976,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_bc,Uu))
    <=> $false ) ).

% ATP.lambda_976
tff(fact_11182_ATP_Olambda__977,axiom,
    ! [Uu: int] :
      ( pp(aa_int_bool(aTP_Lamm_bd,Uu))
    <=> $false ) ).

% ATP.lambda_977
tff(fact_11183_ATP_Olambda__978,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_akq,Uu))
    <=> $true ) ).

% ATP.lambda_978
tff(fact_11184_ATP_Olambda__979,axiom,
    ! [Uu: real] : aa_real_fun_nat_nat(aTP_Lamm_gc,Uu) = suc ).

% ATP.lambda_979
tff(fact_11185_ATP_Olambda__980,axiom,
    ! [Uu: bool] : aa_bool_fun_nat_nat(aTP_Lamm_gd,Uu) = suc ).

% ATP.lambda_980
tff(fact_11186_ATP_Olambda__981,axiom,
    ! [Uu: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ge,Uu) = suc ).

% ATP.lambda_981
tff(fact_11187_ATP_Olambda__982,axiom,
    ! [Uu: int] : aa_int_fun_nat_nat(aTP_Lamm_gf,Uu) = suc ).

% ATP.lambda_982

% Helper facts (106)
tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa_bool_bool(fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,P),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,P),Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,P),Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,P),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,P),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X3: int,Y: int] : aa_int_int(aa_int_fun_int_int(if_int(fFalse),X3),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X3: int,Y: int] : aa_int_int(aa_int_fun_int_int(if_int(fTrue),X3),Y) = X3 ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X3),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X3),Y) = X3 ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y: real] : aa_real_real(if_real(fFalse,X3),Y) = Y ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y: real] : aa_real_real(if_real(fTrue,X3),Y) = X3 ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X3: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X3: bool,Y: bool] :
      ( ( X3 != Y )
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X3: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y: real] :
      ( ( X3 != Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(if_complex(fFalse),X3),Y) = Y ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(if_complex(fTrue),X3),Y) = X3 ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y: complex] :
      ( ( X3 != Y )
      | pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(if_Code_integer(fFalse,X3),Y) = Y ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(if_Code_integer(fTrue,X3),Y) = X3 ).

tff(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X3: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(if_set_int(fFalse),X3),Y) = Y ).

tff(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X3: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(if_set_int(fTrue),X3),Y) = X3 ).

tff(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(if_set_nat(fFalse),X3),Y) = Y ).

tff(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(if_set_nat(fTrue),X3),Y) = X3 ).

tff(help_If_2_1_If_001t__Set__Oset_It__HOL__Obool_J_T,axiom,
    ! [X3: set_bool,Y: set_bool] : if_set_bool(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Set__Oset_It__HOL__Obool_J_T,axiom,
    ! [X3: set_bool,Y: set_bool] : if_set_bool(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X3: vEBT_VEBT,Y: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(fFalse,X3),Y) = Y ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X3: vEBT_VEBT,Y: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(fTrue,X3),Y) = X3 ).

tff(help_If_2_1_If_001t__Set__Oset_It__Real__Oreal_J_T,axiom,
    ! [X3: set_real,Y: set_real] : if_set_real(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Set__Oset_It__Real__Oreal_J_T,axiom,
    ! [X3: set_real,Y: set_real] : if_set_real(fTrue,X3,Y) = X3 ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( ( X3 != Y )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( X3 != Y )
      | pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(aa_set5116768664143533222t_bool(fequal_set_nat,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X3: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( X3 != Y )
      | pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X3: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(aa_VEB2578974996309632030T_bool(fequal_VEBT_VEBT,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Int__Oint_J_T,axiom,
    ! [X3: option_int,Y: option_int] : if_option_int(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Int__Oint_J_T,axiom,
    ! [X3: option_int,Y: option_int] : if_option_int(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X3: option_nat,Y: option_nat] : if_option_nat(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X3: option_nat,Y: option_nat] : if_option_nat(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X3: option_num,Y: option_num] : if_option_num(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X3: option_num,Y: option_num] : if_option_num(fTrue,X3,Y) = X3 ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X3: list_int,Y: list_int] :
      ( ( X3 != Y )
      | pp(aa_list_int_bool(fequal_list_int(X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X3: list_int,Y: list_int] :
      ( ~ pp(aa_list_int_bool(fequal_list_int(X3),Y))
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( X3 != Y )
      | pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ~ pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_If_2_1_If_001t__Option__Ooption_It__HOL__Obool_J_T,axiom,
    ! [X3: option_bool,Y: option_bool] : if_option_bool(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__HOL__Obool_J_T,axiom,
    ! [X3: option_bool,Y: option_bool] : if_option_bool(fTrue,X3,Y) = X3 ).

tff(help_fequal_2_1_fequal_001t__Heap__Oarray_It__Int__Oint_J_T,axiom,
    ! [X3: array_int,Y: array_int] :
      ( ( X3 != Y )
      | pp(fequal_array_int(X3,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Heap__Oarray_It__Int__Oint_J_T,axiom,
    ! [X3: array_int,Y: array_int] :
      ( ~ pp(fequal_array_int(X3,Y))
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Heap__Oarray_It__Nat__Onat_J_T,axiom,
    ! [X3: array_nat,Y: array_nat] :
      ( ( X3 != Y )
      | pp(fequal_array_nat(X3,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Heap__Oarray_It__Nat__Onat_J_T,axiom,
    ! [X3: array_nat,Y: array_nat] :
      ( ~ pp(fequal_array_nat(X3,Y))
      | ( X3 = Y ) ) ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Real__Oreal_J_T,axiom,
    ! [X3: option_real,Y: option_real] : if_option_real(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Real__Oreal_J_T,axiom,
    ! [X3: option_real,Y: option_real] : if_option_real(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Set__Oset_It__Complex__Ocomplex_J_T,axiom,
    ! [X3: set_complex,Y: set_complex] : if_set_complex(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Set__Oset_It__Complex__Ocomplex_J_T,axiom,
    ! [X3: set_complex,Y: set_complex] : if_set_complex(fTrue,X3,Y) = X3 ).

tff(help_fequal_2_1_fequal_001t__Heap__Oarray_It__HOL__Obool_J_T,axiom,
    ! [X3: array_bool,Y: array_bool] :
      ( ( X3 != Y )
      | pp(fequal_array_bool(X3,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Heap__Oarray_It__HOL__Obool_J_T,axiom,
    ! [X3: array_bool,Y: array_bool] :
      ( ~ pp(fequal_array_bool(X3,Y))
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____BuildupMemImp__OVEBTi_T,axiom,
    ! [X3: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( X3 != Y )
      | pp(aa_VEBT_VEBTi_bool(aa_VEB8608361877424424648i_bool(fequal_VEBT_VEBTi,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____BuildupMemImp__OVEBTi_T,axiom,
    ! [X3: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ~ pp(aa_VEBT_VEBTi_bool(aa_VEB8608361877424424648i_bool(fequal_VEBT_VEBTi,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X3: option_nat,Y: option_nat] :
      ( ( X3 != Y )
      | pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X3),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X3: option_nat,Y: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X3),Y))
      | ( X3 = Y ) ) ).

tff(help_If_2_1_If_001t__Set__Oset_It__Code____Numeral__Ointeger_J_T,axiom,
    ! [X3: set_Code_integer,Y: set_Code_integer] : if_set_Code_integer(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Set__Oset_It__Code____Numeral__Ointeger_J_T,axiom,
    ! [X3: set_Code_integer,Y: set_Code_integer] : if_set_Code_integer(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X3: set_VEBT_VEBT,Y: set_VEBT_VEBT] : if_set_VEBT_VEBT(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X3: set_VEBT_VEBT,Y: set_VEBT_VEBT] : if_set_VEBT_VEBT(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X3: heap_Time_Heap_nat,Y: heap_Time_Heap_nat] : if_Hea2662716070787841314ap_nat(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X3: heap_Time_Heap_nat,Y: heap_Time_Heap_nat] : if_Hea2662716070787841314ap_nat(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__HOL__Obool_J_T,axiom,
    ! [X3: heap_Time_Heap_bool,Y: heap_Time_Heap_bool] : if_Hea7696252282596813915p_bool(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__HOL__Obool_J_T,axiom,
    ! [X3: heap_Time_Heap_bool,Y: heap_Time_Heap_bool] : if_Hea7696252282596813915p_bool(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Option__Ooption_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X3: option_VEBT_VEBT,Y: option_VEBT_VEBT] : if_option_VEBT_VEBT(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X3: option_VEBT_VEBT,Y: option_VEBT_VEBT] : if_option_VEBT_VEBT(fTrue,X3,Y) = X3 ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X3: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( X3 != Y )
      | pp(fequal591674592636813711T_VEBT(X3,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X3: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ~ pp(fequal591674592636813711T_VEBT(X3,Y))
      | ( X3 = Y ) ) ).

tff(help_If_2_1_If_001t__Option__Ooption_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X3: option_VEBT_VEBTi,Y: option_VEBT_VEBTi] : if_option_VEBT_VEBTi(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X3: option_VEBT_VEBTi,Y: option_VEBT_VEBTi] : if_option_VEBT_VEBTi(fTrue,X3,Y) = X3 ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X3: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] :
      ( ( X3 != Y )
      | pp(fequal1389750275112723248_VEBTi(X3,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X3: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] :
      ( ~ pp(fequal1389750275112723248_VEBTi(X3,Y))
      | ( X3 = Y ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fFalse),X3),Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fTrue),X3),Y) = X3 ).

tff(help_fequal_2_1_fequal_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X3: array_VEBT_VEBTi,Y: array_VEBT_VEBTi] :
      ( ( X3 != Y )
      | pp(fequal8089995073349378887_VEBTi(X3,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X3: array_VEBT_VEBTi,Y: array_VEBT_VEBTi] :
      ( ~ pp(fequal8089995073349378887_VEBTi(X3,Y))
      | ( X3 = Y ) ) ).

tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X3: heap_T8145700208782473153_VEBTi,Y: heap_T8145700208782473153_VEBTi] : if_Hea8453224502484754311_VEBTi(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X3: heap_T8145700208782473153_VEBTi,Y: heap_T8145700208782473153_VEBTi] : if_Hea8453224502484754311_VEBTi(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_T,axiom,
    ! [X3: heap_T2636463487746394924on_nat,Y: heap_T2636463487746394924on_nat] : if_Hea5867803462524415986on_nat(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_T,axiom,
    ! [X3: heap_T2636463487746394924on_nat,Y: heap_T2636463487746394924on_nat] : if_Hea5867803462524415986on_nat(fTrue,X3,Y) = X3 ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X3: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] : if_opt6109864365331422477at_nat(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X3: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] : if_opt6109864365331422477at_nat(fTrue,X3,Y) = X3 ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X3: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X3,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X3: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X3,Y) = X3 ).

% Conjectures (1)
tff(conj_0,conjecture,
    hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,v),aa_nat_nat(suc,zero_zero_nat),vg),vh)),tia),vEBT_VEBT_vebt_succi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(aa_Pro4773402355715522498at_nat(some_P7363390416028606310at_nat,v),aa_nat_nat(suc,zero_zero_nat),vg),vh),tia,vi),aTP_Lamm_a) ).

%------------------------------------------------------------------------------